#483739

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	C₂³	C₈×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ #2162 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

394 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2)(3,11)(4,8)(5,17)(6,9)(7,13)(10,19)(12,25)(14,24)(15,23)(16,29)(18,21)(20,27)(22,28)(26,32)(30,31)(33,34,38,42,37,44,54,61,39,43,51,46,55,41,58,60)(35,45,49,53,50,57,63,52,47,59,62,64,48,40,56,36);(1,3,6,14)(2,7,9,21)(4,12,15,26)(5,18,19,13)(8,20,23,31)(10,24,17,11)(16,30,28,27)(22,32,29,25)(33,35,39,47)(34,40,43,57)(36,46,52,42)(37,50,55,48)(38,49,51,62)(41,59,44,45)(53,60,64,61)(54,63,58,56);(1,4,6,15)(2,8,9,23)(3,12,14,26)(5,16,19,28)(7,20,21,31)(10,22,17,29)(11,25,24,32)(13,27,18,30)(33,36,51,59,37,53,58,40,39,52,38,45,55,64,54,57)(34,35,46,62,44,50,60,56,43,47,42,49,41,48,61,63);(2,9)(4,15)(7,21)(10,17)(11,24)(12,26)(16,28)(27,30)(35,47)(36,52)(40,57)(45,59)(48,50)(49,62)(53,64)(56,63)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fg,e^2=g,f^2=h,h^2=i,b^a=be,d^a=dgi,e^a=eg,d^b=di,e^b=eg,d^c=dg

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390935);