#7090257

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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×#550	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1671 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)(17,18,23,21)(19,27,29,43)(20,25,32,35)(22,36,37,26)(24,38,34,30)(28,42,31,45)(33,48,44,41)(39,47,40,46);(9,10)(11,15)(12,14)(13,16)(17,19,22,30,23,29,37,38)(18,24,26,27,21,34,36,43)(20,31,33,47,32,28,44,46)(25,40,41,45,35,39,48,42);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20,23,32)(18,25,21,35)(19,28,29,31)(22,33,37,44)(24,39,34,40)(26,41,36,48)(27,42,43,45)(30,46,38,47);(11,16)(13,15)(17,21,23,18)(19,27,29,43)(20,25,32,35)(22,26,37,36)(24,38,34,30)(28,45,31,42)(33,48,44,41)(39,46,40,47);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(18,21)(19,29)(20,32)(26,36)(30,38)(33,44)(39,40)(42,45)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gh,d^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=df,e^b=eh,d^c=dh,e^c=ei,e^d=ei,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6997453);