#72343436

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator	C₂²×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
socle	C₂³	C₂²×#97
prime radical	C₂²×D₄	#97
LCS	G ▷ C₂²×C₄ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

992 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^10|a^2=e,b^2=g,c^2=f,e^2=h,g^2=i,i^2=j,b^a=bij,c^a=cf,d^b=df,d^c=dfhj,e^b=ej,g^a=gj

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343438, #72343570, #72343901
maximal normal subgroup of	#72343438, #72343570, #72343901

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