#490876

order	512
nilpotency class	4
isoclinism	isoclinic class of #490873
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	#3169	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

482 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 138 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,h^a=hi,d^c=dg,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, 398072);