#72287511

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	#10580937	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #3444 ◁ #60204 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

4570 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:

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

Polycyclic presentation:

2^10|c^2=h,d^2=f,e^2=fghi,f^2=h,g^2=i,i^2=j,c^a=cj,c^b=ch,d^a=dhj,d^c=dhj,e^a=ej,e^b=eh,e^c=ej,e^d=eg,f^e=fj,g^d=gi,g^e=gj,i^d=ij

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287517, #72287659, #72319749
maximal normal subgroup of	#72287517, #72287659, #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, 3, F.8);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.6*F.7*F.8*F.9);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.7);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);