#491061

order	512
nilpotency class	4
isoclinism	isoclinic class of #491017
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#33817	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

459 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398257);