#6814078

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1421 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)(11,13)(12,14)(15,16)(17,18,21,23)(19,27,30,24)(20,25,33,35)(22,26,29,36)(28,37,42,39)(31,41,44,38)(32,45,34,40)(43,47,46,48);(1,3,6,2)(4,8,5,7)(9,11,12,15)(10,13,14,16)(18,23)(19,28)(20,32)(22,29)(24,37)(25,40)(27,39)(30,42)(33,34)(35,45)(38,41)(43,46);(10,14)(13,16)(17,19,29,42,21,30,22,28)(18,24,36,37,23,27,26,39)(20,31,32,46,33,44,34,43)(25,38,45,47,35,41,40,48);(1,4,6,5)(2,7,3,8)(18,23)(19,30)(25,35)(26,36)(28,42)(31,44)(40,45)(43,46);(1,5,6,4)(2,7,3,8)(17,20,22,34,21,33,29,32)(18,25,26,40,23,35,36,45)(19,31,28,43,30,44,42,46)(24,38,39,48,27,41,37,47)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=hi,d^2=fi,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, 6721274);