#72307359

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#17595
commutator	C₂×C₈	C₂³×C₈
Frattini, cosocle	C₂×#97	C₂⁴
prime radical	C₂×#3184	C₄
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #20331 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1455 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,5,8)(3,7,4,6)(10,16,21,32)(12,26)(13,27)(15,29)(17,30,20,19)(18,34,37,31)(22,23)(25,39)(28,38)(33,35,36,40);(2,6)(3,4)(7,8)(9,10,13,19,15,20,12,16,14,21,26,30,29,17,27,32)(11,18,23,35,25,36,28,31,24,37,38,40,39,33,22,34);(1,3,5,4)(2,7,8,6)(10,17,21,20)(12,27)(13,26)(16,30,32,19)(18,33,37,36)(22,28)(23,38)(31,40,34,35);(2,8)(6,7)(9,11)(10,18)(12,28)(13,23)(14,24)(15,25)(16,31)(17,33)(19,35)(20,36)(21,37)(22,27)(26,38)(29,39)(30,40)(32,34)

Polycyclic presentation:

2^10|a^2=gj,b^2=f,c^2=gj,e^2=h,f^2=i,h^2=j,i^2=j,b^a=be,c^b=ch,d^a=dgj,e^a=eh,e^b=eg,e^c=ej,e^d=ej,f^a=fgh,f^c=fj,f^e=fj,g^b=gj,h^a=hj,i^a=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307374, #72307684, #72307707
maximal normal subgroup of	#72307374, #72307684, #72307707

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