#141868

order	512
nilpotency class	4
isoclinism	isoclinic class of #13371
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#476
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1987
prime radical	C₂³	#418
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

310 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 102 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=def,c^2=eg,d^2=g,e^2=i,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=ch,d^b=dg,f^b=fgi,g^b=gi,h^b=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49064);