Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25446 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#25448 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2428 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3067 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2592 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)(5,17,31,36,21,35,20,11)(6,22,25,12,27,42,26,39)(7,28,32,41,30,40,29,13)(18,37,46,43,23,33,47,51)(19,48,58,60,50,34,49,44)(24,54,55,52,57,38,56,62)(45,64,65,61,53,63,66,59);(3,5,9,21)(4,11)(6,23)(7,19,30,50)(8,20,10,31)(12,37,39,33)(13,34)(14,17)(15,35)(16,36)(18,27)(22,51,42,43)(24,53)(25,47)(26,46)(28,44)(29,49,32,58)(38,61,54,59)(40,60)(41,48)(45,57)(52,64,62,63)(55,66)(56,65);(3,6)(4,12)(5,18)(7,24)(8,25)(9,27)(10,26)(11,33)(13,38)(14,22)(15,42)(16,39)(17,43)(19,45)(20,46)(21,23)(28,52)(29,55)(30,57)(31,47)(32,56)(34,59)(35,51)(36,37)(40,62)(41,54)(44,63)(48,61)(49,65)(50,53)(58,66)(60,64);(3,7,10,32,9,30,8,29)(4,13,15,40,16,41,14,28)(5,19,31,58,21,50,20,49)(6,24,26,56,27,57,25,55)(11,34,35,60,36,48,17,44)(12,38,42,62,39,54,22,52)(18,45,47,66,23,53,46,65)(33,59,51,64,37,61,43,63)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351590);