#136673

order	512
nilpotency class	3
isoclinism	isoclinic class of #1685
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#121
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1869
prime radical	#3100	C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

386 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 106 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:

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

Polycyclic presentation:

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

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, 43869);