#72322143

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	#32554
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂²×#135	C₂³
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#2961 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1509 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)(5,6,11,16,12,15,13,17,14,18,33,34,10,27,32,7)(8,21,23,35,22,28,36,30,26,9,20,29,24,31,25,19);(2,4)(5,7,10,16,14,17,12,34)(6,11,15,32,18,33,27,13)(8,19,22,29,26,30,24,35)(9,23,28,36,21,20,31,25);(2,4)(5,8,12,22,14,26,10,24)(6,9,27,28,18,21,15,31)(7,19,34,29,17,30,16,35)(11,20,13,25,33,23,32,36);(2,4)(5,9,10,31,14,21,12,28)(6,8,15,24,18,26,27,22)(7,20,16,36,17,23,34,25)(11,19,32,35,33,30,13,29)

Polycyclic presentation:

2^10|a^2=f,b^2=e,c^2=egi,d^2=egij,e^2=j,f^2=i,g^2=j,h^2=ij,i^2=j,b^a=bfgj,c^a=cefgh,c^b=cegj,d^a=defi,d^b=dei,d^c=dj,e^a=ej,e^c=ej,e^d=ej,f^b=fi,f^c=fj,f^d=fi,g^b=gj,g^c=gj,h^b=hij,h^d=hi,i^b=ij,i^d=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322170, #72322623
maximal normal subgroup of	#72322170, #72322623

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