Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188830 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#380 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | C2×C4⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
766 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,10,7,11,5,20)(3,12,16,22,14,26,15,8)(4,9,19,24,17,23,18,30)(13,25,28,31,29,32,27,21)(33,34)(35,43)(36,47,48,40)(37,50)(38,42)(39,45)(41,51)(44,57,58,55)(46,56)(49,54,63,62)(52,59,61,64)(53,60);(1,3,5,15,7,14,6,16)(2,8,10,12,11,22,20,26)(4,13,18,27,17,29,19,28)(9,21,24,25,23,31,30,32)(33,35,37,45,38,46,51,60)(34,39,41,43,42,53,50,56)(36,44,49,59,48,58,63,64)(40,52,54,55,47,61,62,57);(1,4)(2,9)(3,13)(5,18)(6,19)(7,17)(8,21)(10,24)(11,23)(12,25)(14,29)(15,27)(16,28)(20,30)(22,31)(26,32)(33,36,38,48)(34,40,42,47)(35,44,46,58)(37,49,51,63)(39,52,53,61)(41,54,50,62)(43,55,56,57)(45,59,60,64);(3,14)(4,17)(8,22)(9,23)(12,26)(15,16)(18,19)(24,30)(36,48)(40,47)(44,58)(49,63)(52,61)(54,62)(55,57)(59,64)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=i,e^2=h,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dhi,d^c=dh,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 96138);