#6697261

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	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1588 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)(3,6)(4,5)(7,8)(10,14)(11,12)(17,18)(19,24)(20,26)(21,34)(22,25)(23,28)(27,36)(29,43)(30,38)(31,46)(32,40)(33,44)(35,48)(37,41)(39,47)(42,45);(9,10)(11,15)(12,16)(13,14)(17,19,22,32)(18,24,25,40)(20,30,23,29)(21,35,36,45)(26,38,28,43)(27,42,34,48)(31,37,47,33)(39,44,46,41);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(18,25)(19,29)(20,23)(21,36)(24,38)(30,32)(31,45)(35,47)(39,42)(40,43)(41,44)(46,48);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,14,15)(17,20,22,23)(18,26,25,28)(19,30,32,29)(21,33,36,37)(24,38,40,43)(27,41,34,44)(31,45,47,35)(39,48,46,42);(10,14)(15,16)(17,21,22,36)(18,27,25,34)(19,31,32,47)(20,33,23,37)(24,39,40,46)(26,41,28,44)(29,35,30,45)(38,48,43,42)

Polycyclic presentation:

2^9|b^2=i,c^2=f,d^2=g,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, 6604457);