Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24808 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#24810 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2416 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2440 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);(3,4,8,31,10,15,9,14)(5,16,19,11,21,35,20,36)(6,22,32,12,26,39,25,40)(7,27,28,13,30,55,29,41)(17,33,23,37,45,59,44,58)(18,46,47,42,49,34,48,60)(24,38,56,54,53,62,52,50)(43,61,51,65,64,66,63,57);(3,5,8,19,10,21,9,20)(4,11,14,16,15,36,31,35)(6,23,25,17,26,44,32,45)(7,18,28,47,30,49,29,48)(12,37,39,59,40,58,22,33)(13,34,27,42,41,46,55,60)(24,51,52,43,53,63,56,64)(38,61,54,65,62,66,50,57);(3,6)(4,12)(5,17)(7,24)(8,25)(9,32)(10,26)(11,33)(13,38)(14,39)(15,40)(16,37)(18,43)(19,44)(20,23)(21,45)(22,31)(27,54)(28,52)(29,56)(30,53)(34,57)(35,58)(36,59)(41,62)(42,61)(46,65)(47,63)(48,51)(49,64)(50,55)(60,66);(3,7,9,29,10,30,8,28)(4,13,31,55,15,41,14,27)(5,18,20,48,21,49,19,47)(6,24,32,56,26,53,25,52)(11,34,35,60,36,46,16,42)(12,38,22,50,40,62,39,54)(17,43,23,51,45,64,44,63)(33,57,58,66,59,65,37,61)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350255);