#7094473

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1244 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=gh,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=df,e^b=ehi,f^b=fi,d^c=dhi,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, 7001669);