#72308245

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×#1999	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#399 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4893 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)(5,6,9,8)(7,10,11,12)(21,22,25,30)(23,28,26,27)(24,32,34,35)(29,31,36,33);(1,2)(3,4)(5,7)(6,10)(8,12)(9,11)(15,19)(17,20);(2,4)(5,8,9,6)(7,10,11,12)(13,14)(15,20)(16,18)(17,19)(21,23,25,26)(22,27,30,28)(24,33,34,31)(29,32,36,35);(7,11)(10,12)(13,15,16,19)(14,17,18,20)(21,23,25,26)(22,28,30,27)(24,31,34,33)(29,32,36,35);(14,18)(17,20)(21,24,26,33,25,34,23,31)(22,29,28,32,30,36,27,35)

Polycyclic presentation:

2^10|a^2=g,c^2=g,d^2=h,e^2=i,i^2=j,c^a=cf,c^b=cfg,d^b=dgh,d^c=dhj,e^a=ei,e^c=eh,i^a=ij

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308428
maximal normal subgroup of	#72308428

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