#6832819

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806671
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

1271 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)(19,30)(20,25)(21,29)(22,40)(23,28)(24,34)(26,39)(27,36)(31,46)(32,42)(33,43)(35,47)(37,45)(38,44)(41,48);(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,36)(21,32)(22,37)(23,34)(25,40)(26,42)(27,45)(28,30)(29,47)(31,38)(33,48)(35,39)(41,44)(43,46);(2,5)(6,8)(10,14)(11,16)(17,20,21,38,23,37,39,33)(18,25,29,44,28,45,26,43)(19,31,35,36,34,48,32,22)(24,41,42,40,30,46,47,27);(10,14)(13,15)(17,21,23,39)(18,26,28,29)(19,32,34,35)(20,33,37,38)(22,31,36,48)(24,42,30,47)(25,44,45,43)(27,46,40,41);(1,3,4,7)(2,6,5,8)(17,22,39,48,23,36,21,31)(18,27,26,46,28,40,29,41)(19,33,32,37,34,38,35,20)(24,43,47,45,30,44,42,25)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=fhi,d^2=i,e^2=ghi,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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6740015);