#6819708

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806623
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 ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1295 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)(4,8)(5,7)(9,10)(11,14)(12,16)(13,15)(17,18,22,26)(19,23,28,34)(20,24,32,38)(21,25,31,37)(27,33,39,43)(29,35,41,45)(30,36,42,46)(40,44,47,48);(1,3,5,4)(2,6,7,8)(9,11,14,10)(12,16,13,15)(19,27)(20,30)(21,31)(23,33)(24,36)(25,37)(28,39)(29,41)(32,42)(34,43)(35,45)(38,46);(1,4,5,3)(2,6,7,8)(9,12,14,13)(10,15,11,16)(17,19,21,27,22,28,31,39)(18,23,25,33,26,34,37,43)(20,29,30,40,32,41,42,47)(24,35,36,44,38,45,46,48);(9,12,14,13)(10,15,11,16)(19,28)(23,34)(27,39)(29,41)(33,43)(35,45)(40,47)(44,48);(9,13,14,12)(10,15,11,16)(17,20,31,42,22,32,21,30)(18,24,37,46,26,38,25,36)(19,29,39,47,28,41,27,40)(23,35,43,48,34,45,33,44)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^2=f,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6726904);