#72322222

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	#10576895	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #3444 ◁ #60204 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

4688 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:

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

Polycyclic presentation:

2^10|b^2=f,e^2=hj,f^2=h,g^2=ij,i^2=j,b^a=bg,c^a=cj,c^b=cj,d^a=dhj,e^a=eh,e^b=ej,e^c=ej,e^d=eh,f^a=fj,g^a=gi,g^b=gij,i^a=ij,i^b=ij

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322260, #72322669
maximal normal subgroup of	#72322260, #72322669

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