#72257566

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	C₂×C₄	#3184
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
socle	C₂²	#23761
prime radical	C₂⁵	#137
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1487 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^10|b^2=h,c^2=g,d^2=ehj,e^2=ij,g^2=j,i^2=j,c^a=cf,c^b=ch,d^a=dgij,d^b=dgj,d^c=de,e^b=ej,e^c=ei,e^d=ej,f^d=fj,h^d=hj,i^c=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257573, #72257601, #72257702
maximal normal subgroup of	#72257573, #72257601, #72257702

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