#72307103

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#8375
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂×#131	C₂⁴
socle	C₂	C₂×#8375
prime radical	#10576101	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

5828 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^10|a^2=j,c^2=hj,d^2=fg,e^2=j,f^2=ij,i^2=j,b^a=bj,c^a=chj,d^a=df,d^c=de,e^a=ej,e^d=eh,f^a=fi,g^a=gj,g^c=ghj,g^e=gj,h^d=hj,i^a=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307167, #72307535
maximal normal subgroup of	#72307167, #72307535

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