#72264756

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	C₂²×#466	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

5179 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)(16,18)(17,20)(21,22,26,30,25,29,24,28)(23,31,32,36,33,27,34,35);(1,2)(3,4)(5,6,9,12)(7,10,8,11)(15,20)(17,19)(21,23)(22,27)(24,34)(25,33)(26,32)(28,36)(29,31)(30,35);(1,2)(3,4)(7,8)(10,11)(15,20)(17,19);(1,3,4,2)(5,7)(6,10)(8,9)(11,12)(13,15,16,20)(14,17,18,19)(21,24,25,26)(22,28,29,30)(23,32,33,34)(27,35,31,36);(2,3)(5,8,9,7)(6,11,12,10)(14,18)(17,19)(22,28)(24,26)(27,36)(29,30)(31,35)(32,34)

Polycyclic presentation:

2^10|a^2=i,b^2=f,d^2=g,e^2=f,i^2=j,b^a=bj,d^a=dh,d^b=dg,d^c=dfgj,e^a=ehi,e^b=eghj,e^c=efghj,e^d=efgh,i^e=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264861, #72308733
maximal normal subgroup of	#72264861, #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.9);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.6);;
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*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.6*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.6*F.7*F.8);;
gap> SetConjugate(coll, 9, 5, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);