#72250893

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	#23708
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

1653 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:

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

Polycyclic presentation:

2^10|a^2=h,b^2=e,c^2=h,d^2=fhi,e^2=i,g^2=j,i^2=j,b^a=bi,c^b=cegj,d^a=df,d^b=dhi,d^c=dfgi,e^a=ej,e^c=ei,g^c=gj,g^d=gj,h^b=hj,i^c=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250937, #72257203, #72257601
maximal normal subgroup of	#72250937, #72257203, #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.5);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 4, F.6*F.8*F.9);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.7*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.7*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);