#72321920

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	#31681
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂×#3050	C₂²
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20385 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1525 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)(6,12)(7,17,19,22)(8,9)(10,23)(13,30,18,36)(14,16)(20,35,21,26)(24,27)(28,34,31,32)(29,33);(1,3,4,2)(5,6,11,16)(7,18,19,13)(8,14,9,12)(10,25,27,15)(17,34,22,32)(20,31,21,28)(23,29,24,33)(26,30,35,36);(2,3)(7,19)(17,22)(20,21)(26,35);(2,3)(5,7,11,19)(6,13)(8,21,9,20)(10,26,27,35)(12,28)(14,31)(15,30)(16,18)(17,24,22,23)(25,36)(29,32)(33,34)

Polycyclic presentation:

2^10|a^2=e,b^2=fgj,d^2=gij,f^2=j,g^2=j,h^2=i,i^2=j,b^a=bf,c^a=cfg,c^b=cefg,d^a=dfghj,d^b=defg,d^c=dgij,e^d=ej,f^a=fj,f^b=fj,f^d=fj,g^a=gj,g^b=gj,g^d=gj,h^a=hi,h^b=hj,h^d=hij,i^a=ij,i^d=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321926, #72322221
maximal normal subgroup of	#72321926, #72322221

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