#135869

order	512
nilpotency class	3
isoclinism	isoclinic class of #1655
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

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	#1850
prime radical	#121	M₁₆
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

263 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 263 conjugacy classes of subgroups — too many to draw. Only its 88 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,d^2=g,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,d^b=dg,e^b=ei,d^c=di,e^c=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43065);