#7081628

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×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1524 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)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,24,29)(19,30,35,42)(20,36,21,26)(22,28,23,27)(25,33,43,34)(31,47,37,41)(32,40,48,44)(38,45,39,46);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,34,24,35,22,33)(18,25,28,30,29,43,27,42)(20,37,38,32,21,31,39,48)(26,44,45,47,36,40,46,41);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,20)(18,26)(19,31)(21,24)(22,39)(23,38)(25,40)(27,46)(28,45)(29,36)(30,41)(32,33)(34,48)(35,37)(42,47)(43,44);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(17,21,24,20)(18,26,29,36)(19,32,35,48)(22,39,23,38)(25,41,43,47)(27,45,28,46)(30,40,42,44)(31,34,37,33);(10,14)(13,15)(17,22,24,23)(18,27,29,28)(19,33,35,34)(20,38,21,39)(25,42,43,30)(26,45,36,46)(31,48,37,32)(40,41,44,47)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=hi,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,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, 6988824);