Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24420 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C2×#24423 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3586 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4452 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);(3,4,10,32,9,14,8,15)(5,16,30,35,20,11,19,41)(6,21,31,12,24,42,25,38)(7,26,28,39,29,13,27,40)(17,45,51,43,47,36,22,33)(18,34,49,60,50,44,48,59)(23,52,56,57,54,62,55,37)(46,64,53,58,65,61,66,63);(3,5,9,20)(4,11,14,16)(6,22,24,51)(7,18,29,50)(8,30,10,19)(12,36,38,45)(13,34,26,44)(15,41,32,35)(17,31,47,25)(21,33,42,43)(23,53,54,66)(27,49,28,48)(37,61,57,64)(39,60,40,59)(46,55,65,56)(52,58,62,63);(3,6)(4,12)(5,17)(7,23)(8,31)(9,24)(10,25)(11,33)(13,37)(14,38)(15,42)(16,43)(18,46)(19,51)(20,47)(21,32)(22,30)(26,57)(27,56)(28,55)(29,54)(34,58)(35,45)(36,41)(39,52)(40,62)(44,63)(48,53)(49,66)(50,65)(59,61)(60,64);(3,7)(4,13)(5,18)(6,23)(8,28)(9,29)(10,27)(11,34)(12,37)(14,26)(15,40)(16,44)(17,46)(19,49)(20,50)(21,52)(22,53)(24,54)(25,56)(30,48)(31,55)(32,39)(33,58)(35,60)(36,61)(38,57)(41,59)(42,62)(43,63)(45,64)(47,65)(51,66)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345180);