#140142

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=dg,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, 47338);