#72308378

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂³×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
prime radical	C₂×#25421	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#399 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

5181 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^10|a^2=g,b^2=f,d^2=gj,e^2=i,i^2=j,b^a=bf,c^a=cf,c^b=cfg,d^a=df,d^b=dfj,d^c=dh,e^a=ei,e^b=ehj,e^c=eh,e^d=ehj,i^a=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308428, #72308848
maximal normal subgroup of	#72308428, #72308848

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