#72250574

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	#31996
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

1615 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:

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

Polycyclic presentation:

2^10|b^2=h,c^2=hi,d^2=efj,e^2=ij,f^2=j,g^2=j,i^2=j,c^a=ce,c^b=cegj,d^a=degij,d^b=defj,d^c=defgh,e^a=ei,e^b=eij,e^c=eij,e^d=ej,f^a=fj,f^b=fj,f^d=fj,g^a=gj,g^b=gj,g^c=gj,g^d=gj,h^c=hj,h^d=hj,i^a=ij,i^b=ij,i^c=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250612, #72250847, #72257203, …
maximal normal subgroup of	#72250612, #72250847, #72257203, …

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