#140132

order	512
nilpotency class	3
isoclinism	isoclinic class of #118505
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#388
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1924
prime radical	#532	C₂×C₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

247 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 247 conjugacy classes of subgroups — too many to draw. Only its 84 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)(3,5)(4,7)(6,8)(9,10,14,20,13,18,22,17,15,19,31,37,29,38,12,27)(11,21,23,36,26,16,28,34,25,35,30,39,24,33,40,32);(9,11,22,40,29,24,14,30,15,25,12,28,13,26,31,23)(10,16,17,36,38,21,20,32,19,33,27,39,18,35,37,34);(2,5)(3,8)(6,7)(9,12,13,14,15,22,29,31)(10,17,18,37,19,27,38,20)(11,23,24,40,25,30,26,28)(16,32,21,39,33,34,35,36)

Polycyclic presentation:

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

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, 47328);