#140538

order	512
nilpotency class	3
isoclinism	isoclinic class of #1714
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#127
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×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

351 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 351 conjugacy classes of subgroups — too many to draw. Only its 110 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)(9,10,14,18,15,19,12,26,16,20,28,35,13,30,29,40)(11,17,23,33,24,32,27,39,25,34,37,31,22,38,21,36);(2,6,7,8)(9,11,14,23,15,24,12,27,16,25,28,37,13,22,29,21)(10,17,18,33,19,32,26,39,20,34,35,31,30,38,40,36);(2,7)(6,8)(9,12,13,14,16,29,15,28)(10,18,19,26,20,35,30,40)(11,21,22,37,25,27,24,23)(17,31,32,36,34,33,38,39)

Polycyclic presentation:

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

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, 47734);