Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13442 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#13443 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2054 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
998 conjugacy classes of subgroups.
| minimal degree | μ(G) = 66 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 66 points:
(1,2);(3,4,10,16)(5,17,22,52)(6,23,29,59)(7,14,32,49)(8,13,33,48)(9,34,31,15)(11,36,40,19)(12,41,46,26)(18,53,57,37)(20,39,30,50)(21,38,35,47)(24,60,54,65)(25,45,27,63)(28,43,62,44)(42,55,51,64)(56,58,61,66);(3,5)(4,11)(6,24)(7,30)(8,19)(9,35)(10,22)(12,42)(13,47)(14,17)(15,50)(16,40)(18,27)(20,32)(21,31)(23,58)(25,57)(26,56)(28,64)(29,54)(33,36)(34,39)(37,44)(38,48)(41,61)(43,53)(45,60)(46,51)(49,52)(55,62)(59,66)(63,65);(3,6,7,25,31,62,33,41,10,29,32,27,9,28,8,26)(4,12,13,43,34,63,49,59,16,46,48,44,15,45,14,23)(5,18,19,54,35,61,20,55,22,57,36,24,21,56,30,64)(11,37,17,51,50,66,38,65,40,53,52,42,39,58,47,60)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406374);