#72319121

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×#102
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	#10580932	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #3444 ◁ #60204 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

4565 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(2,6)(3,9)(7,12)(11,15)(17,18)(19,24)(20,34)(21,27)(22,26)(23,25)(28,46)(29,48)(30,40)(31,41)(32,42)(33,39)(35,45)(36,44)(37,43)(38,47);(2,7,6,12)(8,14,10,16)(17,19,21,31,22,33,36,28,23,30,37,32,20,29,35,47)(18,24,27,41,26,39,44,46,25,40,43,42,34,48,45,38);(1,2,4,12,5,6,13,7)(3,10,11,14,9,8,15,16)(18,25)(19,28,30,47)(20,22)(21,36)(24,38,40,46)(27,45)(29,32,33,31)(35,37)(39,42,48,41)(43,44);(1,3)(2,8)(4,11)(5,9)(6,10)(7,14)(12,16)(13,15)(17,20,23,22)(18,26,25,34)(19,29,30,33)(21,35,37,36)(24,39,40,48)(27,44,43,45)(28,31,47,32)(38,41,46,42);(3,9)(8,10)(11,15)(14,16)(19,30)(24,40)(28,47)(29,33)(31,32)(38,46)(39,48)(41,42)

Polycyclic presentation:

2^10|b^2=f,c^2=fghj,d^2=j,f^2=h,g^2=i,h^2=j,i^2=j,c^a=chi,c^b=cg,d^a=dhi,d^c=dhi,e^b=ej,e^d=ehij,f^c=fi,g^b=gj,g^c=gi,h^c=hj,i^c=ij

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319144, #72319199, #72319749
maximal normal subgroup of	#72319144, #72319199, #72319749

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