#6447449

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1649 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)(11,14)(12,16)(13,15)(17,18)(19,23)(20,25)(21,24)(22,26)(27,43)(28,37)(29,38)(30,36)(31,41)(32,39)(33,42)(34,40)(35,44)(45,48)(46,47);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,19,21,29)(18,23,24,38)(20,31,32,45)(22,27,34,30)(25,41,39,48)(26,43,40,36)(28,33,46,35)(37,42,47,44);(9,12,13,11)(10,14,15,16)(18,24)(19,27)(22,34)(23,36)(25,39)(28,45)(29,30)(31,46)(33,35)(37,41)(38,43)(47,48);(17,20,21,32)(18,25,24,39)(19,28,29,46)(22,35,34,33)(23,37,38,47)(26,44,40,42)(27,45,30,31)(36,41,43,48);(1,3,4,7)(2,6,5,8)(18,24)(23,38)(25,39)(26,40)(36,43)(37,47)(41,48)(42,44)

Polycyclic presentation:

2^9|b^2=g,c^2=gi,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dhi,h^b=hi,h^c=hi,h^d=hi

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, 6354645);