#72308541

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×#28483	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ #13847 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

5011 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)(5,6,8,10)(7,9,11,12)(15,19)(18,20)(23,29)(25,32)(27,34)(28,33)(30,35)(31,36);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(14,17)(18,20)(21,22)(23,30)(24,26)(25,28)(27,31)(29,35)(32,33)(34,36);(13,14,16,17)(15,20,19,18)(22,26)(27,30)(28,33)(34,35);(2,4)(7,11)(9,12)(13,15)(14,18)(16,19)(17,20)(22,26)(27,30)(28,33)(34,35);(15,19)(18,20)(21,23,25,29,24,31,32,36)(22,27,28,34,26,30,33,35)

Polycyclic presentation:

2^10|a^2=g,b^2=g,c^2=ghj,e^2=i,i^2=j,c^b=cgh,d^a=df,d^b=dfh,d^c=dghj,e^a=ei,e^b=ej,e^d=eghj,i^a=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308569, #72308734
maximal normal subgroup of	#72308569, #72308734

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