#134836

order	512
nilpotency class	4
isoclinism	isoclinic class of #134745
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1807
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#532	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

269 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 269 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2,3,6,4,5,7,8)(9,10,15,19,16,20,14,32)(11,21,23,31,24,30,18,13)(12,25,27,17,29,34,28,35)(22,37,38,40,39,26,33,36);(9,11,13,20,14,23,30,32,16,24,31,10,15,18,21,19)(12,22,26,35,28,38,40,25,29,39,37,17,27,33,36,34);(2,5)(3,7)(9,12)(10,17)(11,22)(13,26)(14,28)(15,27)(16,29)(18,33)(19,34)(20,35)(21,36)(23,38)(24,39)(25,32)(30,40)(31,37)

Polycyclic presentation:

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

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, 42032);