#7085913

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1341 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)(19,31)(20,35)(21,27)(22,29)(23,28)(24,30)(25,33)(26,32)(34,46)(36,47)(37,43)(38,42)(39,45)(40,44)(41,48);(9,10)(11,15)(12,14)(13,16)(17,19,22,40,24,34,23,33)(18,25,28,46,30,44,29,31)(20,36,41,39,32,21,37,38)(26,45,48,47,35,42,43,27);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,20,24,32)(18,26,30,35)(19,21,34,36)(22,41,23,37)(25,42,44,45)(27,31,47,46)(28,48,29,43)(33,39,40,38);(11,16)(13,15)(17,21,24,36)(18,27,30,47)(19,32,34,20)(22,39,23,38)(25,43,44,48)(26,46,35,31)(28,42,29,45)(33,37,40,41);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,22,24,23)(18,28,30,29)(19,33,34,40)(20,37,32,41)(21,38,36,39)(25,31,44,46)(26,43,35,48)(27,45,47,42)

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=h,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=dfi,e^b=eh,d^c=dh,e^c=ei,e^d=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, 6993109);