#6842406

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	#2414	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)(19,23,29,37)(20,30,33,24)(21,25,34,40)(27,35,42,47)(28,39,32,36)(31,44,43,38)(41,48,45,46);(9,11,12,16)(10,13,14,15)(19,27)(20,31)(21,34)(23,35)(24,38)(25,40)(28,32)(29,42)(30,44)(33,43)(36,39)(37,47);(2,6)(7,8)(10,14)(11,16)(17,19,21,27,22,29,34,42)(18,23,25,35,26,37,40,47)(20,32,31,45,33,28,43,41)(24,39,38,48,30,36,44,46);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(20,33)(24,30)(28,32)(31,43)(36,39)(38,44)(41,45)(46,48);(1,4,5,3)(2,8,6,7)(17,20,34,43,22,33,21,31)(18,24,40,44,26,30,25,38)(19,28,42,45,29,32,27,41)(23,36,47,48,37,39,35,46)

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^c=ei,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, 6749602);