#490739

order	512
nilpotency class	4
isoclinism	isoclinic class of #490693
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	C₂×#135	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

481 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 481 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=eg,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,c^b=cg,f^c=fh,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397935);