#135873

order	512
nilpotency class	3
isoclinism	isoclinic class of #1619
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#120
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1850
prime radical	C₂×#41	M₁₆
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

285 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 285 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=g,d^2=g,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,c^b=ci,d^b=dg,e^c=egi

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, 43069);