#106205

order	512
nilpotency class	3
isoclinism	isoclinic class of #106194
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	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

675 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 675 conjugacy classes of subgroups — too many to draw. Only its 180 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) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,6,10,7,12,5,11)(3,13,15,8,16,23,21,24)(4,17,19,9,20,25,18,26)(14,28,29,32,30,22,31,27)(33,34)(35,40,46,55)(36,48)(37,44)(38,43)(39,42)(41,50)(45,59,61,63)(47,57,51,54)(49,58)(52,56)(53,62,64,60);(1,3)(2,8)(4,14)(5,21)(6,15)(7,16)(9,22)(10,23)(11,13)(12,24)(17,32)(18,31)(19,29)(20,30)(25,27)(26,28)(33,35,38,47,39,46,37,51)(34,40,44,54,42,55,43,57)(36,45,50,62,49,61,52,60)(41,53,48,63,56,64,58,59);(1,4,7,20)(2,9,12,26)(3,14,16,30)(5,19,6,18)(8,22,24,28)(10,17,11,25)(13,27,23,32)(15,31,21,29)(33,36,39,49)(34,41,42,56)(35,45,46,61)(37,52,38,50)(40,53,55,64)(43,58,44,48)(47,62,51,60)(54,63,57,59)

Polycyclic presentation:

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

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, 13401);