#72308803

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×#2535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ #13847 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

5011 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)(5,6,8,10)(7,9,11,12)(14,17)(18,20)(23,29)(25,31)(27,34)(28,33)(30,35)(32,36);(1,2)(3,4)(5,7)(6,9)(8,11)(10,12)(15,19)(18,20)(21,22)(23,27)(24,26)(25,28)(29,34)(30,32)(31,33)(35,36);(2,4)(5,7)(6,9)(8,11)(10,12)(13,14)(15,20)(16,17)(18,19)(22,26)(23,30)(28,33)(29,35);(7,11)(9,12)(13,15,16,19)(14,18,17,20)(22,26)(27,32)(28,33)(34,36);(14,17)(18,20)(21,23,25,29,24,30,31,35)(22,27,28,34,26,32,33,36)

Polycyclic presentation:

2^10|a^2=g,d^2=hj,e^2=i,i^2=j,c^a=cf,c^b=cfh,d^b=dgh,d^c=dghj,e^a=ei,e^c=eh,i^a=ij

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308848
maximal normal subgroup of	#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, 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.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*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);