Finite Group Databasebeta

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#72285733

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#2309C23
prime radicalC22×H(5,𝔽2)D4
LCSG ▷ #379 ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #2309 ◁ G
LECSG ▷ #2309 ▷ C24 ▷ C22 ▷ C2 ▷ C1

1225 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 1225 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:

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

Polycyclic presentation:

2^10|a^2=gij,b^2=deg,c^2=gh,d^2=gj,e^2=g,b^a=bd,c^a=cf,c^b=chij,d^a=dg,d^c=dh,e^a=ej,e^c=eh,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#72285762, #72285916
maximal normal subgroup of#72285762, #72285916

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*F.9*F.10);;
gap> SetPower(coll, 2, F.4*F.5*F.7);;
gap> SetPower(coll, 3, F.7*F.8);;
gap> SetPower(coll, 4, F.7*F.10);;
gap> SetPower(coll, 5, F.7);;
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.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
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);