#6813378

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1305 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)(11,13)(12,14)(15,16)(17,18)(19,30)(20,33)(21,36)(22,38)(23,29)(24,27)(25,32)(26,35)(28,40)(31,42)(34,43)(37,44)(39,45)(41,48)(46,47);(2,5)(3,7)(9,11,12,15)(10,13,14,16)(17,19,23,32)(18,25,29,30)(20,24,35,21)(22,39,40,34)(26,27,33,36)(28,45,38,43)(31,41,47,37)(42,44,46,48);(10,14)(13,16)(17,20,24,32,23,35,21,19)(18,26,27,30,29,33,36,25)(22,39,41,31,40,34,37,47)(28,45,44,42,38,43,48,46);(2,5)(6,8)(17,21,23,24)(18,27,29,36)(19,20,32,35)(22,37,40,41)(25,33,30,26)(28,44,38,48)(31,34,47,39)(42,45,46,43);(1,3,4,7)(2,6,5,8)(17,22,24,41,23,40,21,37)(18,28,27,44,29,38,36,48)(19,31,20,34,32,47,35,39)(25,42,26,43,30,46,33,45)

Polycyclic presentation:

2^9|b^2=g,c^2=h,d^2=i,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, 6720574);