#6582684

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573523
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1371 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=gi,c^2=f,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

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, 6489880);