#72252168

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#15670
commutator, Frattini, cosocle	C₂⁴×C₄	C₂⁴
prime radical	C₂⁴×D₄	C₂³
LCS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

5985 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^10|a^2=e,b^2=i,d^2=ej,e^2=j,f^2=j,g^2=j,b^a=befj,c^a=cfi,c^b=ch,d^a=defi,d^b=defgj,d^c=dfij,e^b=ej,e^c=ej,f^a=fj,f^c=fj,f^d=fj,g^a=gj,g^b=gj,g^c=gj,g^d=gj,h^d=hj,i^a=ij,i^d=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252181, #72256010, #72256318
maximal normal subgroup of	#72252181, #72256010, #72256318

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