#6622100

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574298
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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1410 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)(11,16)(12,15)(13,14)(17,18,22,27)(19,28,31,24)(20,32,34,46)(21,26,23,38)(25,35,43,36)(29,44,33,47)(30,41,37,40)(39,45,42,48);(2,5)(6,8)(10,14)(11,12)(17,19,22,31)(18,24,27,28)(20,33,34,29)(21,37,23,30)(25,42,43,39)(26,41,38,40)(32,44,46,47)(35,45,36,48);(9,11,13,12)(10,15,14,16)(17,20,21,35,22,34,23,36)(18,25,38,46,27,43,26,32)(19,29,37,48,31,33,30,45)(24,39,40,44,28,42,41,47);(1,3,4,7)(2,6,5,8)(11,12)(15,16)(20,34)(25,43)(29,33)(32,46)(35,36)(39,42)(44,47)(45,48);(9,12,13,11)(10,15,14,16)(17,21,22,23)(18,26,27,38)(19,30,31,37)(20,35,34,36)(24,40,28,41)(25,32,43,46)(29,45,33,48)(39,44,42,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gi,d^2=fh,e^2=h,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dh,e^d=ehi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6529296);