#72264751

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	#627850	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#2290 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

5059 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:

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

Polycyclic presentation:

2^10|a^2=i,b^2=f,d^2=f,e^2=hij,h^2=j,i^2=j,b^a=bj,d^a=dh,d^b=dg,d^c=dfg,e^a=eh,e^b=eghi,e^c=efghij,e^d=efj,h^d=hj,h^e=hj,i^d=ij,i^e=ij

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264861
maximal normal subgroup of	#72264861

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