#6466733

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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

1397 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)(19,24)(20,25)(21,23)(22,26)(27,39)(28,38)(29,34)(30,42)(31,41)(32,35)(33,36)(37,44)(40,45)(43,48)(46,47);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(18,23)(19,27)(20,30)(22,33)(24,37)(25,40)(28,31)(29,44)(32,45)(34,39)(35,42)(47,48);(9,12,13,11)(10,14,15,16)(17,19,22,27,21,29,33,44)(18,24,26,39,23,34,36,37)(20,31,30,46,32,28,45,43)(25,41,42,47,35,38,40,48);(17,20,22,30,21,32,33,45)(18,25,26,42,23,35,36,40)(19,28,27,43,29,31,44,46)(24,38,39,48,34,41,37,47);(1,3,4,7)(2,6,5,8)(19,29)(24,34)(27,44)(28,31)(37,39)(38,41)(43,46)(47,48)

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^2=h,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,d^c=di,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6373929);