#72308495

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	C₂³×D₈	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ #13856 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4433 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:

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

Polycyclic presentation:

2^10|a^2=f,b^2=fg,c^2=gh,d^2=gh,e^2=i,i^2=j,b^a=bh,c^a=cfgh,c^b=cfg,d^a=dfhj,d^b=dfgj,d^c=dgj,e^a=egij,e^b=egj,e^c=eg,e^d=eg,i^a=ij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308569, #72308733
maximal normal subgroup of	#72308569, #72308733

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