#6829614

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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 ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1267 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)(9,10,12,14)(11,15,16,13)(17,18,21,23)(19,27,30,24)(20,32,35,25)(22,26,29,37)(28,38,43,40)(31,39,34,42)(33,41,36,45)(44,48,46,47);(9,11,12,16)(10,13,14,15)(18,23)(19,28)(20,33)(22,29)(24,38)(25,41)(27,40)(30,43)(32,45)(35,36)(39,42)(44,46);(2,5)(6,8)(10,14)(11,16)(17,19,29,43,21,30,22,28)(18,24,37,38,23,27,26,40)(20,34,33,44,35,31,36,46)(25,42,45,48,32,39,41,47);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(18,23)(19,30)(25,32)(26,37)(28,43)(31,34)(41,45)(44,46);(17,20,22,36,21,35,29,33)(18,25,26,41,23,32,37,45)(19,31,28,44,30,34,43,46)(24,39,40,48,27,42,38,47)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=fh,d^2=fg,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6736810);