#7086010

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

1335 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)(17,18,24,30)(19,31,35,43)(20,26,32,47)(21,27,36,48)(22,29,23,28)(25,33,44,34)(37,42,39,46)(38,41,40,45);(9,10)(11,15)(12,14)(13,16)(17,19,22,33,24,35,23,34)(18,25,28,43,30,44,29,31)(20,36,39,38,32,21,37,40)(26,45,42,27,47,41,46,48);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,20,24,32)(18,26,30,47)(19,21,35,36)(22,39,23,37)(25,41,44,45)(27,43,48,31)(28,42,29,46)(33,40,34,38);(11,16)(13,15)(17,21,24,36)(18,27,30,48)(19,32,35,20)(22,38,23,40)(25,42,44,46)(26,31,47,43)(28,45,29,41)(33,39,34,37);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,22,24,23)(18,28,30,29)(19,33,35,34)(20,37,32,39)(21,38,36,40)(25,43,44,31)(26,46,47,42)(27,45,48,41)

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=h,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=dfi,e^b=ehi,d^c=dh,e^c=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, 6993206);