#483742

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
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

397 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)(33,34,38,43,37,44,54,61,39,41,51,46,55,42,58,60)(35,45,49,52,50,57,63,53,47,59,62,64,48,40,56,36);(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)(34,40,41,57)(36,46,53,43)(37,50,55,48)(38,49,51,62)(42,59,44,45)(52,60,64,61)(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)(33,36,51,59,37,52,58,40,39,53,38,45,55,64,54,57)(34,35,46,62,44,50,60,56,41,47,43,49,42,48,61,63);(2,9)(4,15)(7,21)(10,17)(11,24)(12,26)(16,28)(27,30)(34,41)(35,47)(42,44)(43,46)(48,50)(49,62)(56,63)(60,61)

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390938);