#7086394

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	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1399 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)(17,18,22,28,23,27,39,45)(19,29,32,42,31,47,35,24)(20,33,37,41,36,25,43,44)(21,38,30,48,40,26,34,46);(9,10)(11,15)(12,14)(13,16)(17,19,23,31)(18,24,27,42)(20,34,36,30)(21,37,40,43)(22,35,39,32)(25,38,33,26)(28,47,45,29)(41,48,44,46);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,20,23,36)(18,25,27,33)(19,30,31,34)(21,32,40,35)(22,37,39,43)(24,26,42,38)(28,44,45,41)(29,46,47,48);(11,16)(13,15)(17,21,39,34,23,40,22,30)(18,26,45,48,27,38,28,46)(19,20,35,43,31,36,32,37)(24,41,47,33,42,44,29,25);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(18,27)(19,31)(20,36)(26,38)(28,45)(32,35)(37,43)(46,48)

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,d^a=di,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, 6993590);