#6462713

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1401 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)(11,14)(12,13)(15,16)(17,18,23,24,22,28,21,27)(19,29,36,41,32,25,30,38)(20,26,37,39,35,43,33,44)(31,42,46,48,34,40,45,47);(2,5)(6,8)(18,24)(19,30)(20,33)(21,23)(27,28)(31,34)(32,36)(35,37)(38,41)(39,44)(40,48)(42,47);(10,13)(14,16)(17,19,21,30,22,32,23,36)(18,25,27,41,28,29,24,38)(20,34,33,46,35,31,37,45)(26,42,44,47,43,40,39,48);(1,3,4,7)(2,6,5,8)(17,20,21,33,22,35,23,37)(18,26,27,44,28,43,24,39)(19,31,30,45,32,34,36,46)(25,40,41,48,29,42,38,47);(9,11,12,15)(10,14,13,16)(20,35)(26,43)(31,34)(33,37)(39,44)(40,42)(45,46)(47,48)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6369909);