#72322368

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#31996
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	#411612	C₂
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#2932 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1780 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|a^2=f,b^2=e,c^2=i,d^2=ghi,e^2=j,f^2=i,g^2=ij,h^2=ij,i^2=j,b^a=bfij,c^a=cefgij,c^b=cghi,d^a=dfghi,d^b=dfh,d^c=defhij,e^a=ej,f^b=fi,f^c=fj,f^d=fij,g^b=gij,g^c=gj,g^d=gi,h^b=hij,h^d=hi,i^b=ij,i^d=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322386, #72322646
maximal normal subgroup of	#72322386, #72322646

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