#499707

order	512
nilpotency class	4
isoclinism	isoclinic class of #499210
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#498
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#23837
prime radical	C₂³	#575
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

419 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 419 conjugacy classes of subgroups — too many to draw. Only its 160 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=eh,c^2=e,d^2=ef,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gh,h^b=hi,g^c=gh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406903);