#72239902

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#24338
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂×#3414	C₂²
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20435 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2347 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)(7,16,19,32)(8,10)(9,23)(12,26,28,36)(13,20)(14,29)(17,30,27,33)(18,34,21,31)(22,25)(24,35);(5,6)(7,17,19,27)(8,20)(9,24)(10,13)(11,15)(12,18,28,21)(14,25)(16,30,32,33)(22,29)(23,35)(26,34,36,31);(1,2,3,4)(7,18,19,21)(12,27,28,17)(16,31,32,34)(26,30,36,33);(5,7,9,16,10,18,23,31,11,19,25,32,8,21,22,34)(6,12,14,26,13,17,29,33,15,28,24,36,20,27,35,30)

Polycyclic presentation:

2^10|a^2=h,b^2=h,c^2=g,d^2=f,e^2=j,f^2=i,i^2=j,c^a=cg,c^b=ch,d^a=df,d^b=di,d^c=de,e^a=ej,e^b=ej,e^d=ej,f^a=fi,f^b=fj,g^d=gj,h^d=hj,i^a=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239925
maximal normal subgroup of	#72239925
maximal quotient of	#72239905

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