#6476562

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470402
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1605 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6383758);