#491697

order	512
nilpotency class	4
isoclinism	isoclinic class of #491644
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#19603	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

582 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 174 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=eg,c^2=fhi,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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398893);