#7086138

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1383 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=h,d^2=fi,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=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, 6993334);