#72264909

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	#417634	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#399 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

5129 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)(7,9)(8,10)(11,12)(13,14)(15,17)(16,18)(19,20)(21,22,25,27,26,30,24,29)(23,28,32,34,33,36,31,35);(5,7,8,11)(6,9,10,12)(13,15)(14,17)(16,19)(18,20)(22,27)(24,25)(28,34)(29,30)(31,32)(35,36);(1,2)(3,4)(14,18)(15,19)(21,23,26,33)(22,28,30,36)(24,31,25,32)(27,34,29,35);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(22,29)(24,25)(27,30)(28,35)(31,32)(34,36);(2,4)(6,10)(9,12)(14,18)(17,20)(22,29)(24,25)(27,30)(28,35)(31,32)(34,36)

Polycyclic presentation:

2^10|a^2=i,b^2=f,c^2=j,g^2=j,i^2=j,b^a=bij,c^a=ch,c^b=ch,d^a=di,e^a=efgj,e^c=eghi,e^d=efghi,g^b=gj,g^d=gj,g^e=gj,i^b=ij,i^d=ij,i^e=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265187
maximal normal subgroup of	#72265187

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