Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25798 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
| direct product | C2×#25800 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2433 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3584 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4834 conjugacy classes of subgroups.
| minimal degree | μ(G) = 66 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 66 points:
(1,2);(3,4,9,14)(5,15,19,40)(6,11,23,35)(7,24,27,54)(8,28,25,13)(10,29,32,18)(12,36,37,26)(16,41,20,38)(17,45,47,31)(21,50,52,66)(22,53,42,34)(30,44,57,49)(33,58,59,51)(39,55,61,46)(43,56,48,64)(60,63,62,65);(3,5,8,18,9,19,25,29)(4,10,13,15,14,32,28,40)(6,20,22,49,23,16,42,44)(7,17,26,46,27,47,36,55)(11,30,34,38,35,57,53,41)(12,31,24,39,37,45,54,61)(21,48,51,65,52,43,58,63)(33,56,50,60,59,64,66,62);(3,6)(4,11)(5,16,19,20)(7,21)(8,22)(9,23)(10,30,32,57)(12,33)(13,34)(14,35)(15,38,40,41)(17,43,47,48)(18,44,29,49)(24,50)(25,42)(26,51)(27,52)(28,53)(31,56,45,64)(36,58)(37,59)(39,60,61,62)(46,63,55,65)(54,66);(3,7,25,36,9,27,8,26)(4,12,28,54,14,37,13,24)(5,17,29,55,19,47,18,46)(6,21,42,58,23,52,22,51)(10,31,40,61,32,45,15,39)(11,33,53,66,35,59,34,50)(16,43,49,65,20,48,44,63)(30,56,41,62,57,64,38,60)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350536);