#129082

order	512
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄×C₈	C₂×D₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₄×C₄⋊C₈
prime radical	C₂³	#431
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

276 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 146 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=de,d^2=g,f^2=h,h^2=i,b^a=bd,c^a=cgi,d^b=dg,e^b=eg

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36278);