#7085866

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

1359 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)(17,18,23,27)(19,29,32,45)(20,34,31,25)(21,26,35,46)(22,38,30,28)(24,40,43,33)(36,42,47,48)(37,44,39,41);(9,10)(11,15)(12,14)(13,16)(17,19,30,40,23,32,22,33)(18,24,38,45,27,43,28,29)(20,35,47,37,31,21,36,39)(25,44,48,46,34,41,42,26);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,20,23,31)(18,25,27,34)(19,21,32,35)(22,36,30,47)(24,41,43,44)(26,29,46,45)(28,42,38,48)(33,37,40,39);(11,16)(13,15)(17,21,23,35)(18,26,27,46)(19,31,32,20)(22,39,30,37)(24,42,43,48)(25,45,34,29)(28,44,38,41)(33,36,40,47);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(18,27)(19,32)(20,31)(26,46)(28,38)(33,40)(36,47)(41,44)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=h,d^2=i,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,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, 6993062);