#72250705

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#21526
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂³×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20436 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1568 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)(6,11,16,15)(8,21)(12,26,23,28)(13,22)(14,27,32,33)(18,24)(29,35,31,36)(30,34);(2,4)(6,12,16,23)(7,17)(8,22)(9,25)(11,26,15,28)(13,21)(14,29,32,31)(18,34)(24,30)(27,35,33,36);(5,6,13,28,25,15,21,12,10,16,22,26,9,11,8,23)(7,14,30,36,19,33,18,29,17,32,34,35,20,27,24,31);(1,2,3,4)(5,7,9,20,10,17,25,19)(6,14,11,27,16,32,15,33)(8,24,22,34,21,18,13,30)(12,29,28,36,23,31,26,35)

Polycyclic presentation:

2^10|a^2=h,b^2=h,c^2=egj,d^2=fi,e^2=i,g^2=j,i^2=j,c^a=ci,c^b=ce,d^a=df,d^b=dfgij,e^a=ej,e^b=ei,e^c=ej,e^d=ej,g^b=gj,g^c=gj,g^d=gj,h^c=hj,i^b=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250744, #72257186, #72257573
maximal normal subgroup of	#72250744, #72257186, #72257573

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