#72322721

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#24338
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂²×#509	C₂²
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20438 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1911 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:

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

Polycyclic presentation:

2^10|a^2=h,b^2=e,c^2=efhij,d^2=eghij,e^2=j,f^2=i,g^2=ij,h^2=ij,i^2=j,b^a=bf,c^a=cfhj,c^b=cefgij,d^a=dfhij,d^b=dghij,d^c=degh,e^a=ej,e^c=ej,f^b=fi,f^c=fj,f^d=fj,g^b=gij,g^c=gj,g^d=gj,h^b=hij,h^d=hj,i^b=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322750, #72322854
maximal normal subgroup of	#72322750, #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.8);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.5*F.6*F.8*F.9*F.10);;
gap> SetPower(coll, 4, F.5*F.7*F.8*F.9*F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 8, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.8*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
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.9*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);