Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #27987 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#27988 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2504 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1774 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);(3,4,10,14,9,16,8,15)(5,17,21,34,22,41,20,11)(6,23,25,38,26,37,30,12)(7,27,31,40,29,39,28,13)(18,35,45,42,47,32,46,59)(19,48,50,61,51,33,49,43)(24,36,53,52,54,60,56,55)(44,63,64,58,66,62,65,57);(3,5,8,20,9,22,10,21)(4,11,14,17,16,34,15,41)(6,18,25,45,26,47,30,46)(7,19,28,49,29,51,31,50)(12,35,37,59,38,32,23,42)(13,33,27,43,40,48,39,61)(24,44,53,64,54,66,56,65)(36,58,55,63,60,57,52,62);(3,6,9,26)(4,12)(5,18)(7,24,29,54)(8,25,10,30)(11,32,34,35)(13,36)(14,37)(15,23)(16,38)(17,42,41,59)(19,44)(20,45)(21,46)(22,47)(27,55)(28,53,31,56)(33,57,48,58)(39,52)(40,60)(43,62,61,63)(49,64)(50,65)(51,66);(3,7,10,31,9,29,8,28)(4,13,15,39,16,40,14,27)(5,19,21,50,22,51,20,49)(6,24,30,56,26,54,25,53)(11,33,41,61,34,48,17,43)(12,36,23,52,38,60,37,55)(18,44,46,65,47,66,45,64)(32,57,59,63,35,58,42,62)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6365727);