#7085993

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1354 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)(19,31,35,44)(20,36,32,26)(21,27,37,47)(22,29,23,28)(25,33,45,34)(38,43,40,48)(39,46,41,42);(9,10)(11,15)(12,14)(13,16)(17,19,22,33,24,35,23,34)(18,25,28,44,30,45,29,31)(20,37,40,39,32,21,38,41)(26,46,48,47,36,42,43,27);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,20,24,32)(18,26,30,36)(19,21,35,37)(22,40,23,38)(25,42,45,46)(27,31,47,44)(28,48,29,43)(33,41,34,39);(11,16)(13,15)(17,21,24,37)(18,27,30,47)(19,32,35,20)(22,39,23,41)(25,43,45,48)(26,44,36,31)(28,42,29,46)(33,40,34,38);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,22,24,23)(18,28,30,29)(19,33,35,34)(20,38,32,40)(21,39,37,41)(25,44,45,31)(26,43,36,48)(27,42,47,46)

Polycyclic presentation:

2^9|a^2=g,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=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, 6993189);