#105763

order	512
nilpotency class	3
isoclinism	isoclinic class of #105752
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

940 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 940 conjugacy classes of subgroups — too many to draw. Only its 190 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,d^b=di,e^b=ei,f^b=fh,d^c=di,f^c=fh

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, 12959);