#483369

order	512
nilpotency class	3
isoclinism	isoclinic class of #3186
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	C₂²×C₄⋊C₈
prime radical	C₂⁴	C₄⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

502 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,10)(4,8)(5,14)(6,9)(7,13)(11,16)(12,15)(17,18,21,27,23,26,30,32,22,25,29,24,20,31,19,28);(1,3,5,13,6,11,4,12)(2,7,8,10,9,15,14,16)(17,19,20,29,22,30,23,21)(18,24,26,28,25,27,31,32);(3,11)(7,15)(10,16)(12,13)(18,25)(24,27)(26,31)(28,32);(1,4,6,5)(2,8,9,14)(3,12,11,13)(7,10,15,16)(17,20,22,23)(18,26,25,31)(19,29,30,21)(24,28,27,32)

Polycyclic presentation:

2^9|a^2=f,b^2=e,d^2=g,e^2=g,f^2=h,h^2=i,b^a=be,c^a=ci,d^a=di,e^a=eg,c^b=cgi

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, 390565);