#6580901

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1363 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)(3,7)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,18,22,26)(19,23,30,39)(20,31,33,24)(21,25,34,41)(27,36,28,37)(29,45,46,38)(32,42,35,40)(43,48,44,47);(2,6)(7,8)(9,11,12,15)(10,13,14,16)(19,27)(20,32)(21,34)(23,36)(24,40)(25,41)(28,30)(31,42)(33,35)(37,39)(43,44)(47,48);(1,3,5,4)(2,7,6,8)(10,14)(13,16)(17,19,22,30)(18,23,26,39)(20,29,33,46)(21,28,34,27)(24,38,31,45)(25,37,41,36)(32,43,35,44)(40,47,42,48);(1,4,5,3)(2,8,6,7)(19,28)(20,32)(21,34)(23,37)(24,40)(25,41)(27,30)(29,46)(31,42)(33,35)(36,39)(38,45);(17,20,22,33)(18,24,26,31)(19,29,30,46)(21,35,34,32)(23,38,39,45)(25,42,41,40)(27,43,28,44)(36,47,37,48)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6488097);