#72265321

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	#7623305	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ C₂×#1752 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3177 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^10|a^2=f,b^2=f,d^2=g,g^2=i,i^2=j,c^a=ch,c^b=cj,d^a=di,d^b=dg,e^d=eh,f^d=fj,g^a=gj,g^b=gi,i^b=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265340, #72265397, #72308363
maximal normal subgroup of	#72265340, #72265397, #72308363

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