Finite Group Databasebeta

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#72285535

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#139624
commutator#379C22×C4
Frattini, cosocle#2308C23
prime radicalC22×H(5,𝔽2)D4
LCSG ▷ #379 ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C25 ◁ #2308 ◁ G
LECSG ▷ #2308 ▷ C24 ▷ C22 ▷ C2 ▷ C1

1286 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 1286 conjugacy classes of subgroups — too many to draw. Only its 73 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 64
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2)(3,13)(4,19,23,9)(5,10)(6,26)(7,12)(8,16)(11,28)(14,36)(15,39,20,51)(17,40)(18,41)(21,34,54,50)(22,35,27,38)(24,29)(25,37)(30,44,33,46)(31,47)(32,48)(42,55)(43,59,52,57)(45,61,53,62)(49,56)(58,64,60,63);(1,3,14,42)(2,8,29,56)(4,20,52,63)(5,16,28,49)(6,17,25,48)(7,18,24,47)(9,33,59,61)(10,13,26,55)(11,31,37,41)(12,32,36,40)(15,43,64,23)(19,30,57,62)(21,46,27,45)(22,53,54,44)(34,39,38,58)(35,60,50,51);(1,4,7,23)(2,9,12,19)(3,15)(5,21)(6,27)(8,30)(10,34)(11,38)(13,39,41,51)(14,43,24,52)(16,46,48,44)(17,45,49,53)(18,20)(22,28)(25,54)(26,35)(29,57,36,59)(31,58,55,60)(32,33)(37,50)(40,62)(42,63)(47,64)(56,61)

Polycyclic presentation:

2^10|a^2=g,b^2=degj,c^2=ghj,d^2=g,e^2=gj,b^a=bd,c^a=cf,c^b=chj,d^a=dg,d^b=dij,d^c=dh,e^a=eij,e^b=eij,e^c=ehj,e^d=ej,f^a=fj,f^b=fhj,f^c=fi,f^d=fij,f^e=fi,g^c=gj,h^a=hi,h^b=hj,h^c=hj,h^d=hj,h^e=hj,i^b=ij

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72285575, #72285612, #72341135, …
maximal normal subgroup of#72285575, #72285612, #72341135, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.7);;
gap> SetPower(coll, 2, F.4*F.5*F.7*F.10);;
gap> SetPower(coll, 3, F.7*F.8*F.10);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.7*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);