#72250163

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#32659
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂³×D₈	C₂³
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#2961 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1713 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)(3,4)(5,6,10,15,11,12,9,25)(7,16,17,26,20,27,19,13)(8,21,23,30,22,14,24,31)(18,35,34,29,36,32,33,28);(7,17,20,19)(8,22)(13,27,26,16)(14,21)(18,33,36,34)(23,24)(28,35,29,32)(30,31);(2,4)(6,12)(7,18,20,36)(8,23)(9,10)(13,28,26,29)(14,30)(16,32,27,35)(17,33,19,34)(21,31)(22,24);(5,7,8,18,10,19,23,33,11,20,22,36,9,17,24,34)(6,13,14,29,15,27,31,35,12,26,21,28,25,16,30,32)

Polycyclic presentation:

2^10|a^2=h,b^2=hij,c^2=hij,d^2=e,e^2=i,g^2=j,h^2=j,i^2=j,b^a=bhij,c^a=cf,d^a=dij,d^b=dg,d^c=de,e^a=ej,e^b=ej,e^c=ei,f^a=fj,g^a=gj,g^c=gj,h^c=hj,h^d=hj,i^c=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250220, #72257330, #72257613, …
maximal normal subgroup of	#72250220, #72257330, #72257613, …

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