#72322451

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#31681
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂²×#135	C₂³
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20435 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1351 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:

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

Polycyclic presentation:

2^10|a^2=e,b^2=fgij,d^2=fgi,f^2=i,g^2=i,h^2=i,i^2=j,b^a=bf,c^a=ch,c^b=cij,d^a=dfgi,d^b=deghi,d^c=dghi,e^b=ej,e^c=ej,f^a=fi,f^b=fij,f^c=fij,g^a=gi,g^b=gij,g^c=gij,h^a=hi,h^b=hi,h^c=hij,i^a=ij,i^b=ij,i^c=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322466, #72322623
maximal normal subgroup of	#72322466, #72322623

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