#6688143

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1527 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)(9,10,12,14)(11,15,16,13)(18,24)(25,38)(26,28)(27,39)(29,42)(34,41)(40,45)(43,48);(1,3)(2,5)(4,8)(6,7)(17,18,22,24)(19,29,32,42)(20,28,23,26)(21,34,36,41)(25,37,38,30)(27,33,39,35)(31,45,47,40)(43,46,48,44);(9,11,12,16)(10,13,14,15)(17,19)(18,25)(20,30)(21,31)(22,32)(23,37)(24,38)(26,29)(27,40)(28,42)(33,44)(34,48)(35,46)(36,47)(39,45)(41,43);(2,6)(5,7)(17,20,22,23)(18,26,24,28)(19,30,32,37)(21,35,36,33)(25,29,38,42)(27,34,39,41)(31,46,47,44)(40,48,45,43);(3,8)(5,7)(17,21,22,36)(18,27,24,39)(19,31,32,47)(20,33,23,35)(25,40,38,45)(26,41,28,34)(29,43,42,48)(30,44,37,46)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,d^2=i,e^2=i,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, 6595339);