#72343472

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#442
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	#13304	C₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #8144 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

965 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^10|a^2=e,c^2=h,d^2=f,e^2=h,g^2=i,i^2=j,b^a=bij,c^a=ch,c^b=cf,d^a=dfgij,d^b=dgij,d^c=dfh,e^d=ej,g^a=gi,g^b=gij,g^d=gij,i^a=ij,i^b=ij,i^d=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343487, #72343670, #72343778
maximal normal subgroup of	#72343487, #72343670, #72343778

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