#6832658

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1292 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)(9,10,12,14)(11,15,16,13)(17,18)(19,29)(20,25)(21,27)(22,26)(23,28)(24,31)(30,45)(32,44)(33,41)(34,40)(35,42)(36,43)(37,39)(38,47)(46,48);(9,11)(10,13)(12,16)(14,15)(17,19,22,31)(18,24,26,29)(20,32,33,34)(21,23,37,36)(25,40,41,44)(27,43,39,28)(30,46,47,35)(38,48,45,42);(2,5)(6,8)(10,14)(11,16)(17,20,23,35,22,33,36,46)(18,25,28,42,26,41,43,48)(19,30,21,32,31,47,37,34)(24,38,39,40,29,45,27,44);(10,14)(13,15)(18,26)(20,33)(24,29)(27,39)(28,43)(30,47)(32,34)(35,46);(1,3,4,7)(2,6,5,8)(17,21,36,19,22,37,23,31)(18,27,43,29,26,39,28,24)(20,34,46,47,33,32,35,30)(25,40,48,38,41,44,42,45)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=fhi,e^2=ghi,h^2=i,b^a=bf,c^a=cg,d^a=df,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, 6739854);