#72250087

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	#24033
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	#6442781	C₂
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#2929 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3634 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:

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

Polycyclic presentation:

2^10|c^2=f,d^2=fghi,f^2=ij,g^2=ij,i^2=j,b^a=bh,c^a=ce,c^b=cg,d^a=dfghi,d^b=dh,d^c=de,e^d=ej,f^b=fi,g^b=gi,g^c=gj,i^b=ij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250103, #72257183, #72257271
maximal normal subgroup of	#72250103, #72257183, #72257271

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