#6819050

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	#2397	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1382 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^2=f,e^2=h,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, 6726246);