#483740

order	512
nilpotency class	3
isoclinism	isoclinic class of #3198
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

386 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)(33,34,38,44,40,43,56,62,39,45,49,60,37,53,55,63)(35,46,48,52,47,41,57,50,42,59,61,51,54,58,64,36);(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)(33,35,39,42)(34,41,45,58)(36,44,50,60)(37,54,40,47)(38,48,49,61)(43,59,53,46)(51,63,52,62)(55,64,56,57);(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)(33,36,49,59,40,52,55,58,39,50,38,46,37,51,56,41)(34,42,60,48,43,54,63,57,45,35,44,61,53,47,62,64);(2,9)(4,15)(7,21)(10,17)(11,24)(12,26)(16,28)(27,30)(35,42)(36,50)(41,58)(46,59)(47,54)(48,61)(51,52)(57,64)

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,c^a=ci,d^a=dgi,e^a=eg,d^b=di,e^b=eg,d^c=dg

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390936);