#6822209

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

1306 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,d^2=g,e^2=h,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, 6729405);