#140155

order	512
nilpotency class	3
isoclinism	isoclinic class of #2018
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#121
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1923
prime radical	C₂³	#431
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

231 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 231 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 256

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

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

Polycyclic presentation:

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

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, 47351);