#507688

order	512
nilpotency class	4
isoclinism	isoclinic class of #501040
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

576 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 162 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)(3,7)(4,5)(6,8)(11,15)(13,16)(17,18,24,28,20,27,31,32,21,19,22,26,23,25,29,30);(2,5)(6,7)(9,10,12,14)(11,16,15,13)(18,19)(25,27)(26,28)(30,32);(9,11,12,15)(10,13,14,16)(17,19,21,18)(20,27,23,25)(22,30,24,32)(26,31,28,29);(1,3,4,8)(2,6,5,7)(17,20,21,23)(18,25,19,27)(22,29,24,31)(26,32,28,30)

Polycyclic presentation:

2^9|a^2=fg,b^2=gi,c^2=g,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fh,h^c=hi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414884);