#6718520

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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1627 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)(4,6)(5,8)(9,10,12,13)(11,14,15,16)(17,18,22,27)(19,28,32,24)(20,25,35,43)(21,26,38,23)(29,45,48,41)(30,31,37,39)(33,46,47,42)(34,36,44,40);(1,3,4,8)(2,5,6,7)(10,13)(11,15)(17,19,22,32)(18,24,27,28)(20,33,35,47)(21,37,38,30)(23,39,26,31)(25,42,43,46)(29,40,48,36)(34,45,44,41);(10,13)(14,16)(17,20,22,35)(18,25,27,43)(19,29,32,48)(21,34,38,44)(23,40,26,36)(24,41,28,45)(30,46,37,42)(31,47,39,33);(2,6)(5,7)(10,13)(14,16)(17,21)(18,26)(19,30)(20,34)(22,38)(23,27)(24,39)(25,36)(28,31)(29,46)(32,37)(33,41)(35,44)(40,43)(42,48)(45,47);(3,8)(5,7)(9,11,12,15)(10,14,13,16)(19,31)(21,38)(23,26)(24,30)(28,37)(29,47)(32,39)(33,48)(34,44)(36,40)(41,46)(42,45)

Polycyclic presentation:

2^9|a^2=gi,b^2=fg,c^2=i,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,e^d=ei,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, 6625716);