#72322563

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#31681
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	#60450	C₂²
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20383 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1397 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:

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

Polycyclic presentation:

2^10|a^2=e,b^2=j,d^2=gi,f^2=j,g^2=j,h^2=i,i^2=j,b^a=bf,c^a=ch,c^b=cfgj,d^a=dfghi,d^b=de,d^c=dfj,e^c=ej,e^d=ej,f^a=fj,f^b=fj,f^c=fj,f^d=fj,g^a=gj,g^b=gj,g^c=gj,g^d=gj,h^a=hi,h^b=hj,h^c=hij,h^d=hij,i^a=ij,i^c=ij,i^d=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322569, #72322646
maximal normal subgroup of	#72322569, #72322646

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