#106093

order	512
nilpotency class	3
isoclinism	isoclinic class of #106089
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	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

962 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 962 conjugacy classes of subgroups — too many to draw. Only its 194 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)(11,19,22,34)(12,16)(13,17)(14,18)(15,21,28,39)(20,35,37,33)(23,29)(24,30)(25,31)(26,32)(27,36,40,38);(2,5)(3,7)(9,11)(10,15,18,28)(12,20)(13,21)(14,22)(16,27,30,40)(17,19,32,34)(23,36)(24,37)(25,38)(26,39)(29,35,31,33);(2,5)(3,7)(9,12,14,24)(10,16,18,30)(11,20,22,37)(13,25,26,23)(15,27,28,40)(17,31,32,29)(19,33,34,35)(21,38,39,36)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,e^b=ei,f^b=fhi,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, 13289);