#7081904

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=h,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=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, 6989100);