Finite Group Databasebeta

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#72341131

order1024
nilpotency class6
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#150393
commutator#379C22×C4
Frattini, cosocle#1781C23
prime radicalC22×H(5,𝔽2)D4
LCSG ▷ #379 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #20585 ◁ G
LECSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

1070 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 1070 conjugacy classes of subgroups — too many to draw. Only its 55 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:

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

Polycyclic presentation:

2^10|a^2=e,b^2=i,c^2=fj,d^2=fj,e^2=fh,b^a=bd,c^a=cfghij,c^b=cj,d^a=dg,d^b=df,e^b=efgj,e^c=ehj,e^d=eh,f^a=fi,f^e=fj,g^a=gh,g^c=gij,g^d=gi,g^e=gi,h^a=hi,h^b=hij,h^c=hj,h^d=hj,h^e=hj,i^a=ij

Conjugacy classes & character table

43 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72341132, #72341140, #72341153
maximal normal subgroup of#72341132, #72341140, #72341153

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.5);;
gap> SetPower(coll, 2, F.9);;
gap> SetPower(coll, 3, F.6*F.10);;
gap> SetPower(coll, 4, F.6*F.10);;
gap> SetPower(coll, 5, F.6*F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.7*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*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, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);