#72322141

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#30396
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂×#24288	C₂
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#2932 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3136 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)(5,6,11,16,12,15,34,7,13,17,14,20,10,23,32,21)(8,22,25,30,27,36,33,9,28,18,26,29,24,35,19,31);(2,4)(5,7,10,20,13,21,12,16)(6,14,15,34,17,11,23,32)(8,22,24,36,28,18,27,35)(9,19,29,26,31,33,30,25);(5,8,10,27,13,28,12,24)(6,9,23,29,17,31,15,30)(7,18,16,35,21,22,20,36)(11,33,32,26,14,19,34,25);(5,9,10,30,13,31,12,29)(6,8,23,24,17,28,15,27)(7,19,16,26,21,33,20,25)(11,22,32,35,14,18,34,36)

Polycyclic presentation:

2^10|a^2=f,b^2=e,c^2=gj,d^2=ij,e^2=j,f^2=i,g^2=j,h^2=i,i^2=j,b^a=bfgj,c^a=cehij,c^b=cegj,d^a=deg,d^b=dei,d^c=dgij,e^a=ej,e^c=ej,e^d=ej,f^b=fi,f^c=fi,f^d=fj,g^b=gj,g^d=gj,h^b=hi,h^c=hij,i^b=ij,i^c=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322170, #72322591
maximal normal subgroup of	#72322170, #72322591

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