#140794

order	512
nilpotency class	3
isoclinism	isoclinic class of #113356
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1955
prime radical	C₂²×D₄	M₁₆
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 302 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47990);