#129654

order	512
nilpotency class	3
isoclinism	isoclinic class of #1619
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#120
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1779
prime radical	#3483	C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

619 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 619 conjugacy classes of subgroups — too many to draw. Only its 120 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,d^2=g,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=ci,d^b=dg,e^b=eg,d^c=dg,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36850);