#72265160

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	C₂²×#447	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#399 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

5395 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^10|a^2=f,b^2=f,e^2=fhij,h^2=j,i^2=j,b^a=bh,c^b=chij,d^a=di,d^b=dj,e^a=eij,e^b=efhi,e^c=efg,e^d=efghi,h^a=hj,h^b=hj,h^d=hj,h^e=hj,i^a=ij,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.6);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 5, F.6*F.8*F.9*F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.8*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.6*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*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);