#136759

order	512
nilpotency class	3
isoclinism	isoclinic class of #1655
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

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	#1869
prime radical	#3171	C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 320 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) = 256

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

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

Polycyclic presentation:

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

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, 43955);