#7086806

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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

1308 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)(3,6)(4,8)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,23,28)(19,25,31,41)(20,26,34,42)(21,27,22,24)(29,39,33,43)(30,37,35,40)(32,44,36,38)(45,48,46,47);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,14)(13,16)(18,24)(20,32)(21,22)(25,37)(26,42)(27,28)(29,45)(30,35)(33,46)(34,36)(39,43)(40,41);(2,7)(6,8)(9,11,12,16)(10,13,14,15)(17,19,23,31)(18,25,28,41)(20,33,34,29)(21,35,22,30)(24,37,27,40)(26,43,42,39)(32,46,36,45)(38,48,44,47);(11,16)(13,15)(17,20,23,34)(18,26,28,42)(19,29,31,33)(21,32,22,36)(24,38,27,44)(25,39,41,43)(30,46,35,45)(37,47,40,48);(1,4,5,3)(2,8,7,6)(10,14)(13,15)(17,21,23,22)(18,27,28,24)(19,30,31,35)(20,32,34,36)(25,37,41,40)(26,44,42,38)(29,46,33,45)(39,47,43,48)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=hi,d^2=i,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=ehi,f^b=fi,d^c=dhi,e^c=ei,f^d=fi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6994002);