#72240847

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#23764
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂²×#472	C₂²
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20384 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1574 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^10|a^2=h,b^2=fhi,c^2=h,d^2=egij,e^2=i,i^2=j,b^a=bf,c^b=cfg,d^a=dfi,d^b=dgh,d^c=de,e^a=ej,e^b=ej,e^c=ei,e^d=ej,g^b=gj,g^d=gj,h^d=hj,i^c=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240864, #72257324, #72257601
maximal normal subgroup of	#72240864, #72257324, #72257601

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