#490559

order	512
nilpotency class	4
isoclinism	isoclinic class of #490559
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1677
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#2213	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

448 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 136 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) = 32

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

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

Polycyclic presentation:

2^9|b^2=eh,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,g^a=gi,h^a=hi,d^b=di,f^b=fi,e^c=ei,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, 397755);