Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #28182 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#28184 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2535 |
| 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 | |
2264 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);(3,4,10,16)(5,17,21,35)(6,12,26,39)(7,27,30,40)(8,15,9,14)(11,22,34,20)(13,29,41,28)(18,44,46,36)(19,47,49,33)(23,52,51,32)(24,54,56,62)(25,42,31,38)(37,57,61,55)(43,50,59,48)(45,58,64,65)(53,63,66,60);(3,5,9,22,10,21,8,20)(4,11,15,35,16,34,14,17)(6,23,25,46,26,51,31,18)(7,19,29,50,30,49,28,48)(12,36,38,52,39,44,42,32)(13,33,40,59,41,47,27,43)(24,53,55,64,56,66,57,45)(37,60,54,65,61,63,62,58);(3,6)(4,12)(5,18)(7,24)(8,25)(9,31)(10,26)(11,32)(13,37)(14,38)(15,42)(16,39)(17,36)(19,45)(20,23)(21,46)(22,51)(27,54)(28,55)(29,57)(30,56)(33,58)(34,52)(35,44)(40,62)(41,61)(43,60)(47,65)(48,53)(49,64)(50,66)(59,63);(3,7,9,29,10,30,8,28)(4,13,15,40,16,41,14,27)(5,19,22,50,21,49,20,48)(6,24,31,57,26,56,25,55)(11,33,35,59,34,47,17,43)(12,37,42,62,39,61,38,54)(18,45,51,66,46,64,23,53)(32,58,44,63,52,65,36,60)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6370106);