#72317474

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#9497
commutator	C₂²×#41	C₂²×C₄
Frattini, prime radical, cosocle	C₂³×#41	C₂³
socle	C₂	#120014
LCS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×#41 ▷ C₂³ ▷ C₂ ▷ C₁

1692 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^10|a^2=d,b^2=f,c^2=fi,f^2=j,b^a=bde,c^a=cfi,c^b=cfg,d^b=dh,e^a=eh,e^b=eh,e^c=ei,g^a=gi,g^b=gj,g^d=gj,h^c=hj,i^a=ij,i^b=ij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317506, #72317554, #72317737
maximal normal subgroup of	#72317506, #72317554, #72317737

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