#6722713

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

1380 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)(3,8)(4,6)(5,7)(9,10,13,15)(11,16,12,14)(17,18)(19,23)(20,30)(21,25)(22,26)(24,32)(27,45)(28,40)(29,39)(31,48)(33,44)(34,43)(35,38)(36,42)(37,46)(41,47);(9,10)(11,14)(12,16)(13,15)(17,19,21,29)(18,23,25,39)(20,31,32,47)(22,35,34,28)(24,41,30,48)(26,38,43,40)(27,36,46,33)(37,44,45,42);(1,3,5,4)(2,6,7,8)(9,11,13,12)(10,14,15,16)(17,20,21,32)(18,24,25,30)(19,27,29,46)(22,36,34,33)(23,37,39,45)(26,44,43,42)(28,47,35,31)(38,41,40,48);(9,11,13,12)(10,14,15,16)(20,32)(24,30)(27,46)(31,47)(33,36)(37,45)(41,48)(42,44);(1,4,5,3)(2,6,7,8)(9,12,13,11)(10,14,15,16)(18,25)(19,28)(22,34)(23,38)(24,30)(27,47)(29,35)(31,46)(33,36)(37,41)(39,40)(45,48)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=f,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efhi,h^b=hi,d^c=di,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, 6629909);