#7086027

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×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1382 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)(17,18,23,28)(19,29,33,40)(20,34,36,25)(21,27,22,26)(24,32,41,31)(30,46,35,45)(37,43,38,44)(39,48,42,47);(9,10)(11,15)(12,14)(13,16)(17,19,21,32,23,33,22,31)(18,24,26,40,28,41,27,29)(20,35,38,48,36,30,37,47)(25,42,44,45,34,39,43,46);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,30)(21,38)(22,37)(23,36)(24,39)(26,44)(27,43)(28,34)(29,45)(31,48)(32,47)(33,35)(40,46)(41,42);(11,16)(13,15)(19,31)(20,36)(21,22)(24,29)(25,34)(26,27)(30,47)(32,33)(35,48)(39,46)(40,41)(42,45);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,23,22)(18,26,28,27)(19,32,33,31)(20,37,36,38)(24,40,41,29)(25,43,34,44)(30,48,35,47)(39,45,42,46)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=hi,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,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, 6993223);