#72250091

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#24248
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20385 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2880 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)(11,23)(16,31)(24,36)(28,32);(1,2)(3,4)(7,15)(8,17)(9,21)(11,24)(13,26)(14,27)(16,28)(18,29)(19,35)(20,33)(22,34)(23,36)(25,30)(31,32);(1,3,4,2)(5,6)(7,16)(8,20)(9,14)(10,12)(11,25)(13,22)(15,28)(17,33)(18,31)(19,23)(21,27)(24,30)(26,34)(29,32)(35,36);(5,7,17,30,9,19,22,29,10,18,34,35,21,25,8,15)(6,11,26,32,14,16,20,36,12,23,33,28,27,31,13,24)

Polycyclic presentation:

2^10|c^2=h,d^2=gi,g^2=ij,i^2=j,b^a=bh,c^a=cf,c^b=ch,d^a=de,d^b=dg,d^c=deij,e^c=ej,f^d=fj,g^b=gi,g^c=gj,i^b=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250103, #72257171, #72257284
maximal normal subgroup of	#72250103, #72257171, #72257284

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