#72321931

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#24248
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	#6443262	C₂
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20383 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3529 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^10|a^2=e,b^2=ehij,c^2=ehj,d^2=fi,e^2=ij,f^2=j,g^2=j,h^2=i,i^2=j,b^a=bh,c^a=cij,c^b=cf,d^a=dfgj,d^b=dgi,d^c=defhij,e^b=ei,e^c=ej,f^b=fj,g^b=gj,g^c=gj,h^a=hj,h^b=hi,h^c=hj,i^b=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321960, #72322854
maximal normal subgroup of	#72321960, #72322854

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