#6694865

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1457 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)(11,14)(12,13)(15,16)(17,18)(19,25)(20,34)(21,27)(22,28)(23,29)(24,30)(26,37)(31,47)(32,41)(33,42)(35,46)(36,45)(38,43)(39,48)(40,44);(1,3)(2,5)(4,8)(6,7)(17,19,23,33)(18,25,29,42)(20,35,37,38)(21,32,24,39)(22,36,31,40)(26,43,34,46)(27,41,30,48)(28,45,47,44);(10,13)(14,16)(17,20,23,37)(18,26,29,34)(19,31,33,22)(21,36,24,40)(25,28,42,47)(27,44,30,45)(32,38,39,35)(41,46,48,43);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,21,23,24)(18,27,29,30)(19,32,33,39)(20,36,37,40)(22,35,31,38)(25,41,42,48)(26,44,34,45)(28,46,47,43);(3,8)(5,7)(17,22,23,31)(18,28,29,47)(19,20,33,37)(21,38,24,35)(25,34,42,26)(27,43,30,46)(32,40,39,36)(41,44,48,45)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,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, 6602061);