#72322016

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#32554
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂²×#509	C₂²
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20436 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1422 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)(5,6,11,17,12,16,31,33,13,18,32,7,10,8,14,20)(9,22,26,21,28,23,30,35,29,34,27,24,25,19,36,15);(2,4)(5,7,10,20,13,17,12,33)(6,14,16,32,18,31,8,11)(9,23,25,34,29,19,28,22)(15,27,21,30,35,26,24,36);(1,2)(3,4)(5,8)(6,10)(7,11)(9,24)(12,18)(13,16)(14,20)(15,25)(17,32)(19,36)(21,29)(22,27)(23,30)(26,34)(28,35)(31,33);(2,4)(5,9,10,28,13,29,12,25)(6,15,16,24,18,35,8,21)(7,19,20,34,17,23,33,22)(11,30,14,27,32,36,31,26)

Polycyclic presentation:

2^10|a^2=f,b^2=e,d^2=egij,e^2=j,f^2=i,g^2=j,h^2=i,i^2=j,b^a=bfgj,c^a=cf,c^b=cfhij,d^a=defij,d^b=dei,d^c=dfgh,e^a=ej,e^d=ej,f^b=fi,f^c=fij,f^d=fi,g^b=gj,g^c=gj,h^b=hi,h^c=hij,h^d=hij,i^b=ij,i^c=ij,i^d=ij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322065, #72322386, #72322662
maximal normal subgroup of	#72322065, #72322386, #72322662

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