#72320716

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#24196
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂×#3155	C₂²
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20438 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1870 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:

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

Polycyclic presentation:

2^10|a^2=h,b^2=e,c^2=ei,d^2=efgj,e^2=j,h^2=i,i^2=j,b^a=bhij,c^a=cf,c^b=cf,d^a=dfij,d^b=dg,d^c=defgi,e^a=ej,f^c=fj,g^c=gj,h^b=hi,h^d=hj,i^b=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320850, #72322854, #72323041
maximal normal subgroup of	#72320850, #72322854, #72323041

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