#72252660

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#24089
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂⁵	C₂×SD₁₆
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20385 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1855 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:

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

Polycyclic presentation:

2^10|a^2=f,b^2=h,c^2=fg,h^2=i,i^2=j,b^a=bh,c^a=cfg,c^b=cj,d^b=de,d^c=dg,e^c=ej,f^b=fj,g^b=gj,h^a=hi,i^a=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252664, #72252672, #72252912
maximal normal subgroup of	#72252664, #72252672, #72252912

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