#72250554

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	#31996
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂²×#472	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20384 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1705 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:

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

Polycyclic presentation:

2^10|b^2=h,c^2=gi,d^2=eh,e^2=ij,i^2=j,c^a=cg,c^b=cf,d^a=de,d^b=degij,d^c=dej,e^a=ei,e^b=eij,e^c=ei,e^d=ej,f^c=fj,f^d=fj,g^c=gj,h^d=hj,i^a=ij,i^b=ij,i^c=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250612, #72257311, #72257499, …
maximal normal subgroup of	#72250612, #72257311, #72257499, …

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