Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #356055 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22⋊C42 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2189 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,7,11,26,36)(3,13,17,39,16,8,27,30)(4,18,20,37,21,9,25,32)(5,10,23,35,24,34,51,59)(14,28,42,53,41,38,44,55)(15,33,43,58,22,29,49,54)(19,45,47,60,48,31,50,57)(40,52,61,63,46,56,62,64);(1,3)(2,8,11,13)(4,14)(5,22,24,15)(6,17)(7,16)(9,28,18,38)(10,33)(12,30,36,39)(19,46,48,40)(20,42)(21,41)(23,49,51,43)(25,44)(26,27)(29,34)(31,56)(32,53,37,55)(35,58)(45,52)(47,62,50,61)(54,59)(57,64)(60,63);(1,4,7,21)(2,9,11,18)(3,14,16,41)(5,19,24,48)(6,25,26,20)(8,28,13,38)(10,31,34,45)(12,37,36,32)(15,40,22,46)(17,44,27,42)(23,50,51,47)(29,52,33,56)(30,55,39,53)(35,60,59,57)(43,62,49,61)(54,64,58,63);(1,5)(2,10)(3,15)(4,19)(6,23)(7,24)(8,29)(9,31)(11,34)(12,35)(13,33)(14,40)(16,22)(17,43)(18,45)(20,47)(21,48)(25,50)(26,51)(27,49)(28,52)(30,54)(32,57)(36,59)(37,60)(38,56)(39,58)(41,46)(42,61)(44,62)(53,63)(55,64)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=i,g^2=i,b^a=be,c^a=cf,f^a=fi,h^a=hi,d^b=de,e^b=ei,g^c=gi,h^d=hi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263263);