#72308519

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	#417582	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ #13847 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4898 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)(5,6,9,10)(7,12,8,11)(23,29)(25,32)(27,35)(28,34)(30,36)(31,33);(1,3,4,2)(6,10)(11,12)(13,14,17,16)(15,18,19,20)(21,22,24,26)(23,30,31,27)(25,28,32,34)(29,36,33,35);(2,3)(5,7,9,8)(6,11,10,12)(14,16)(15,19)(22,26)(23,31)(28,34)(29,33);(2,3)(6,10)(11,12)(13,15,17,19)(14,18,16,20)(21,23,32,29,24,31,25,33)(22,27,34,35,26,30,28,36);(5,8,9,7)(6,11,10,12)(13,16,17,14)(15,18,19,20)(22,26)(23,31)(28,34)(29,33)

Polycyclic presentation:

2^10|a^2=g,b^2=fh,c^2=g,d^2=ghi,e^2=hj,i^2=j,b^a=bf,c^a=cf,c^b=cfh,d^a=dfij,d^b=dfgj,d^c=dgh,e^b=ej,e^c=eghj,e^d=egh,i^a=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308569, #72308848
maximal normal subgroup of	#72308569, #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*F.8);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 4, F.7*F.8*F.9);;
gap> SetPower(coll, 5, F.8*F.10);;
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.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);