#490749

order	512
nilpotency class	4
isoclinism	isoclinic class of #12483
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#459
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×#459
prime radical	#33660	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

488 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)(19,22)(20,23)(21,29)(24,28)(25,27)(26,30)(31,32);(1,3,5,13,6,11,4,12)(2,7,8,10,9,15,14,16)(18,22)(23,30)(24,31)(25,29);(3,11)(7,15)(10,16)(12,13)(18,23)(20,28)(21,27)(22,30)(24,25)(26,32)(29,31);(1,4,6,5)(2,8,9,14)(3,12,11,13)(7,10,15,16)

Polycyclic presentation:

2^9|b^2=ei,d^2=g,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,h^a=hi,c^b=cg,f^c=fh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397945);