#140789

order	512
nilpotency class	3
isoclinism	isoclinic class of #10012
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1955
prime radical	#3100	C₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

345 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 345 conjugacy classes of subgroups — too many to draw. Only its 90 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:

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

Polycyclic presentation:

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

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, 47985);