#6699409

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

1413 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=fg,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,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, 6606605);