#6842342

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1303 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)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18,22,26)(19,23,29,36)(20,30,32,24)(21,25,33,38)(27,34,40,46)(28,41,42,35)(31,43,44,37)(39,47,48,45);(9,11,12,16)(10,13,14,15)(19,27)(20,31)(21,33)(23,34)(24,37)(25,38)(28,42)(29,40)(30,43)(32,44)(35,41)(36,46);(2,6)(7,8)(10,14)(11,16)(17,19,21,27,22,29,33,40)(18,23,25,34,26,36,38,46)(20,28,31,39,32,42,44,48)(24,35,37,45,30,41,43,47);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(20,32)(24,30)(28,42)(31,44)(35,41)(37,43)(39,48)(45,47);(1,4,5,3)(2,8,6,7)(17,20,33,44,22,32,21,31)(18,24,38,43,26,30,25,37)(19,28,40,48,29,42,27,39)(23,35,46,47,36,41,34,45)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fh,d^2=fg,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,e^d=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, 6749538);