#6829070

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

1273 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)(19,25)(20,34)(21,27)(22,37)(23,29)(24,30)(26,31)(28,39)(32,47)(33,42)(35,43)(36,46)(38,45)(40,44)(41,48);(9,11)(10,13)(12,16)(14,15)(17,19,23,33)(18,25,29,42)(20,21,31,24)(22,38,39,35)(26,30,34,27)(28,43,37,45)(32,36,48,40)(41,44,47,46);(2,5)(6,8)(10,14)(11,16)(17,20,24,19,23,31,21,33)(18,26,30,42,29,34,27,25)(22,38,40,48,39,35,36,32)(28,45,46,41,37,43,44,47);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,23,24)(18,27,29,30)(19,31,33,20)(22,36,39,40)(25,26,42,34)(28,44,37,46)(32,38,48,35)(41,43,47,45);(17,22,21,36,23,39,24,40)(18,28,27,44,29,37,30,46)(19,32,20,35,33,48,31,38)(25,41,34,45,42,47,26,43)

Polycyclic presentation:

2^9|a^2=f,b^2=i,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, 6736266);