#133872

order	512
nilpotency class	4
isoclinism	isoclinic class of #133847
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#535 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 940 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)(3,7,8,5)(4,6)(9,10)(11,20)(12,15)(13,27,28,18)(14,19)(16,22,31,38)(17,24)(21,34)(23,36)(25,35)(26,33,39,30)(29,40,32,37);(1,3,4,8)(2,5,6,7)(10,15)(12,24)(13,28)(16,29)(17,19)(18,33)(20,34)(21,25)(22,38)(27,30)(31,32)(35,36);(9,11,14,23)(10,16,19,31)(12,25,24,21)(13,22,28,38)(15,29,17,32)(18,20,27,36)(26,40,39,37)(30,35,33,34)

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=i,d^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,d^b=di,e^b=ei,f^b=fgi,d^c=di,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, 41068);