#6687951

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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

1528 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);(1,3)(2,5)(4,8)(6,7)(17,18,23,28)(19,29,32,42)(20,27,22,25)(21,34,36,40)(24,37,39,30)(26,35,41,33)(31,38,46,48)(43,44,47,45);(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,30)(21,31)(22,37)(23,32)(25,29)(26,38)(27,42)(28,39)(33,44)(34,43)(35,45)(36,46)(40,47)(41,48);(2,6)(5,7)(17,20,23,22)(18,25,28,27)(19,30,32,37)(21,35,36,33)(24,29,39,42)(26,40,41,34)(31,45,46,44)(38,47,48,43);(3,8)(5,7)(17,21,23,36)(18,26,28,41)(19,31,32,46)(20,33,22,35)(24,38,39,48)(25,34,27,40)(29,43,42,47)(30,44,37,45)

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=df,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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6595147);