#6622243

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

1400 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)(9,10,13,14)(11,16,12,15)(17,18)(19,24)(20,31)(21,26)(22,36)(23,27)(25,32)(28,43)(29,38)(30,39)(33,41)(34,40)(35,42)(37,46)(44,48)(45,47);(2,5)(6,8)(10,14)(11,12)(17,19,23,29)(18,24,27,38)(20,28,33,44)(21,34,22,30)(25,37,42,47)(26,40,36,39)(31,43,41,48)(32,46,35,45);(9,11,13,12)(10,15,14,16)(17,20,22,32,23,33,21,35)(18,25,26,31,27,42,36,41)(19,28,30,46,29,44,34,45)(24,37,40,43,38,47,39,48);(1,3,4,7)(2,6,5,8)(11,12)(15,16)(19,29)(24,38)(28,44)(30,34)(37,47)(39,40)(43,48)(45,46);(9,12,13,11)(10,15,14,16)(17,21,23,22)(18,26,27,36)(19,30,29,34)(20,32,33,35)(24,39,38,40)(25,41,42,31)(28,45,44,46)(37,43,47,48)

Polycyclic presentation:

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

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, 6529439);