#137858

order	512
nilpotency class	3
isoclinism	isoclinic class of #9239
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1869
prime radical	C₂²×D₄	C₂×C₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

320 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 92 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)(3,6)(4,12,15,10)(5,11)(7,9)(8,14,16,13)(17,18,21,26,23,27,31,30,22,25,28,32,20,19,24,29);(1,3,6,2,7,11,5,9)(4,10,13,8,15,12,14,16)(17,19,28,29,23,18,24,26,22,27,21,30,20,25,31,32);(1,4,7,15)(2,8,9,16)(3,10,11,12)(5,14,6,13)(17,20,22,23)(18,19,25,27)(21,24,28,31)(26,29,32,30)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=g,d^2=gi,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=di,e^b=egi,f^b=fi,d^c=dgi,e^c=egi,f^c=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45054);