#72264567

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×#2856	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

5127 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)(15,17)(16,18)(19,20)(21,22,24,29,25,26,32,28)(23,27,31,34,30,33,36,35);(1,2)(3,4)(5,6,8,10)(7,9,11,12)(15,19)(17,20)(22,26)(27,33)(28,29)(34,35);(1,2)(3,4)(5,7)(6,9)(8,11)(10,12)(15,19)(17,20)(21,23)(22,27)(24,31)(25,30)(26,33)(28,35)(29,34)(32,36);(1,2)(3,4)(13,15,16,19)(14,17,18,20)(22,28)(23,30)(24,32)(26,29)(27,34)(33,35);(2,4)(7,11)(9,12)(14,18)(17,20)(22,29)(24,32)(26,28)(27,34)(31,36)(33,35)

Polycyclic presentation:

2^10|a^2=i,b^2=f,d^2=h,i^2=j,b^a=bj,d^a=di,d^b=dh,d^c=dhj,e^a=ehij,e^b=eg,e^c=efg,e^d=eg,i^d=ij,i^e=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264601
maximal normal subgroup of	#72264601

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