Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24295 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C2×#24296 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2408 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4398 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)(5,16)(6,21)(7,27)(8,31)(9,14)(10,15)(11,19)(12,25)(13,28)(17,33)(18,46)(20,35)(22,37)(23,38)(24,40)(26,39)(29,41)(30,55)(32,36)(34,48)(42,47)(43,61)(44,58)(45,59)(49,51)(50,65)(52,54)(53,62)(56,60)(57,63)(64,66);(3,5,8,19,10,20,9,32)(4,11,14,16,15,36,31,35)(6,22,24,17,26,44,25,45)(7,18,28,47,30,48,29,56)(12,37,39,59,40,58,21,33)(13,34,27,42,41,46,55,60)(23,50,51,43,53,63,52,64)(38,61,54,65,62,66,49,57);(3,6)(4,12)(5,17)(7,23)(8,24)(9,25)(10,26)(11,33)(13,38)(14,39)(15,40)(16,37)(18,43)(19,44)(20,45)(21,31)(22,32)(27,54)(28,51)(29,52)(30,53)(34,57)(35,58)(36,59)(41,62)(42,61)(46,65)(47,63)(48,64)(49,55)(50,56)(60,66);(3,7,9,29,10,30,8,28)(4,13,31,55,15,41,14,27)(5,18,32,56,20,48,19,47)(6,23,25,52,26,53,24,51)(11,34,35,60,36,46,16,42)(12,38,21,49,40,62,39,54)(17,43,22,50,45,64,44,63)(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, 6349894);