#490497

order	512
nilpotency class	4
isoclinism	isoclinic class of #490428
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

498 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 154 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)(3,9,12,16)(4,13,10,7)(6,14,15,11)(18,21)(19,24)(20,23)(25,26)(27,32)(28,31)(29,30);(1,3,10,14,5,12,4,11)(2,6,13,16,8,15,7,9)(17,18,23,26,22,25,20,21)(19,24,30,32,28,31,27,29);(17,19)(18,24)(20,27)(21,29)(22,28)(23,30)(25,31)(26,32);(17,20,22,23)(18,21,25,26)(19,27,28,30)(24,29,31,32)

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,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, 397693);