#134097

order	512
nilpotency class	4
isoclinism	isoclinic class of #133889
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂×#535	C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#536 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 887 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41293);