#72264693

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	#420549	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ #13847 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

5065 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=g,c^2=fg,d^2=hj,e^2=i,i^2=j,c^a=cf,c^b=cfg,d^a=dgh,d^c=dj,e^a=ej,e^b=ei,e^c=eh,i^b=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264843, #72308848
maximal normal subgroup of	#72264843, #72308848

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