#134565

order	512
nilpotency class	4
isoclinism	isoclinic class of #124650
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1527
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
socle	C₂	#8581
prime radical	#532	C₂×C₄
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂×C₄ ▷ C₂ ▷ C₁

294 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 32

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dg,e^2=h,f^2=g,g^2=i,b^a=bd,c^a=ce,e^a=eg,h^a=hi,e^b=ei,d^c=di,e^c=eh,f^c=fghi,g^c=gi,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41761);