#72321970

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#21408
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂×#3060	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20384 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1910 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:

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

Polycyclic presentation:

2^10|a^2=h,b^2=e,c^2=j,d^2=efghj,e^2=j,g^2=i,h^2=ij,i^2=j,b^a=bg,c^a=cf,c^b=cefghij,d^a=dfi,d^b=deghi,d^c=df,e^a=ej,e^c=ej,e^d=ej,g^b=gi,g^c=gj,h^b=hij,h^d=hj,i^b=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322059, #72322277, #72323088
maximal normal subgroup of	#72322059, #72322277, #72323088

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