#140540

order	512
nilpotency class	3
isoclinism	isoclinic class of #9755
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1954
prime radical	#532	C₂×C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 347 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47736);