#141452

order	512
nilpotency class	4
isoclinism	isoclinic class of #141451
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1970
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

360 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2)(3,5,7,8)(4,6)(9,10)(11,22)(12,28)(13,31)(14,32)(15,21)(16,24)(17,25)(18,30)(19,23)(20,29)(26,40)(27,33)(34,37)(35,39)(36,38);(1,3,4,7)(2,5)(6,8)(9,11,23,37,14,26,12,24,15,27,30,39,29,38,13,25)(10,16,31,40,20,34,18,22,21,17,28,36,32,35,19,33);(2,6)(5,8)(9,11,23,37,14,26,12,24,15,27,30,39,29,38,13,25)(10,17,31,36,20,35,18,33,21,16,28,40,32,34,19,22)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=eg,d^2=ghi,e^2=ghi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gi,d^b=dh,f^b=fh

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, 48648);