#6813429

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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

1307 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)(21,39)(22,40)(23,29)(24,27)(25,32)(26,37)(28,41)(31,42)(34,43)(35,44)(36,45)(38,48)(46,47);(2,5)(3,7)(9,11,12,15)(10,13,14,16)(17,19,23,32)(18,25,29,30)(20,34,37,36)(21,22,24,41)(26,43,33,45)(27,40,39,28)(31,35,47,38)(42,48,46,44);(10,14)(13,16)(17,20,21,35,23,37,24,38)(18,26,39,48,29,33,27,44)(19,31,41,36,32,47,22,34)(25,42,40,45,30,46,28,43);(2,5)(6,8)(17,21,23,24)(18,27,29,39)(19,22,32,41)(20,35,37,38)(25,40,30,28)(26,44,33,48)(31,34,47,36)(42,45,46,43);(1,3,4,7)(2,6,5,8)(17,22,21,19,23,41,24,32)(18,28,39,25,29,40,27,30)(20,36,35,47,37,34,38,31)(26,45,48,46,33,43,44,42)

Polycyclic presentation:

2^9|b^2=g,c^2=hi,d^2=i,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^c=ei,e^d=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, 6720625);