#72250294

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#2054
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂³×#509	C₂
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ #20385 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

5653 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);(5,6,9,13,20,15,31,25,11,7,16,18,10,14,26,30)(8,19,21,35,22,17,33,34,24,29,32,36,23,12,27,28);(2,4)(5,7,11,6)(8,19,24,29)(9,25,16,30)(10,14,20,15)(12,23,17,22)(13,26,18,31)(21,28,32,34)(27,35,33,36);(1,2,3,4)(5,8,20,22,11,24,10,23)(6,12,14,29,7,17,15,19)(9,21,31,33,16,32,26,27)(13,28,30,36,18,34,25,35)

Polycyclic presentation:

2^10|b^2=e,c^2=j,d^2=fhj,e^2=ij,g^2=j,h^2=j,i^2=j,c^b=cej,d^a=df,d^b=dh,d^c=dfgj,e^c=ei,g^c=gj,g^d=gj,h^b=hj,i^c=ij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250312, #72250807, #72257728
maximal normal subgroup of	#72250312, #72250807, #72257728

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