#6678805

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1466 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)(3,5)(4,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18,22,26)(19,23,28,38)(20,30,33,24)(21,25,34,42)(27,37,35,43)(29,41,32,39)(31,44,36,40)(45,48,46,47);(1,3,4,7)(2,5,6,8)(10,13)(11,15)(19,27)(20,31)(21,34)(23,37)(24,40)(25,42)(28,35)(30,44)(33,36)(38,43)(45,46)(47,48);(2,6)(5,8)(9,11,12,15)(10,14,13,16)(17,19)(18,23)(20,32)(21,35)(22,28)(24,41)(25,43)(26,38)(27,34)(29,33)(30,39)(31,46)(36,45)(37,42)(40,48)(44,47);(10,13)(14,16)(19,28)(23,38)(27,35)(29,32)(37,43)(39,41)(45,46)(47,48);(17,20,22,33)(18,24,26,30)(19,29,28,32)(21,36,34,31)(23,39,38,41)(25,44,42,40)(27,45,35,46)(37,47,43,48)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,h^e=hi

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, 6586001);