Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #22455 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
| direct product | C2×#22460 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#578 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4464 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);(3,4,9,14,10,15,8,28)(5,16,19,11,20,35,29,34)(6,21,23,37,24,12,31,38)(7,13,26,42,27,41,30,40)(17,45,48,32,49,43,47,52)(18,44,50,33,51,39,25,58)(22,53,56,62,57,36,55,61)(46,60,54,59,64,63,65,66);(3,5)(4,11)(6,17)(7,25)(8,29)(9,19)(10,20)(12,32)(13,39)(14,35)(15,34)(16,28)(18,26)(21,52)(22,54)(23,48)(24,49)(27,50)(30,51)(31,47)(33,40)(36,60)(37,45)(38,43)(41,44)(42,58)(46,55)(53,63)(56,64)(57,65)(59,61)(62,66);(3,6,10,24)(4,12,15,21)(5,17,20,49)(7,22,27,57)(8,23,9,31)(11,32,34,52)(13,36,41,53)(14,37,28,38)(16,43,35,45)(18,46,51,64)(19,47,29,48)(25,54,50,65)(26,55,30,56)(33,59,58,66)(39,60,44,63)(40,61,42,62);(3,7)(4,13)(5,18)(6,22)(8,30)(9,26)(10,27)(11,33)(12,36)(14,42)(15,41)(16,44)(17,46)(19,50)(20,51)(21,53)(23,56)(24,57)(25,29)(28,40)(31,55)(32,59)(34,58)(35,39)(37,62)(38,61)(43,63)(45,60)(47,65)(48,54)(49,64)(52,66)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6286385);