#6580455

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573523
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1529 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)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,22,27)(19,29,32,24)(20,25,37,43)(21,38,35,26)(23,28,36,45)(30,44,48,40)(31,41,34,47)(33,46,39,42);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,19,22,32)(18,24,27,29)(20,34,37,31)(21,30,35,48)(23,39,36,33)(25,41,43,47)(26,44,38,40)(28,46,45,42);(10,14)(13,16)(17,20,22,37)(18,25,27,43)(19,30,32,48)(21,23,35,36)(24,40,29,44)(26,45,38,28)(31,39,34,33)(41,42,47,46);(1,4,5,3)(2,6,7,8)(20,35)(21,37)(23,36)(25,26)(28,45)(30,31)(33,39)(34,48)(38,43)(40,47)(41,44)(42,46);(17,21,22,35)(18,26,27,38)(19,31,32,34)(20,36,37,23)(24,41,29,47)(25,28,43,45)(30,33,48,39)(40,46,44,42)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=i,d^2=fi,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,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6487651);