#72250709

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

1564 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:

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

Polycyclic presentation:

2^10|a^2=h,c^2=fgi,d^2=eghj,e^2=i,i^2=j,c^a=cg,c^b=cf,d^a=de,d^b=deij,e^a=ei,e^b=eij,e^c=ej,f^c=fj,f^d=fj,g^c=gj,g^d=gj,h^d=hj,i^a=ij,i^b=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250744, #72257230, #72257613
maximal normal subgroup of	#72250744, #72257230, #72257613

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