#134627

order	512
nilpotency class	4
isoclinism	isoclinic class of #134444
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1802
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#417
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 300 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) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dgi,c^2=g,d^2=i,e^2=hi,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,e^c=eh,f^c=fgh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41823);