Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13005 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#13006 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1874 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1342 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);(3,4,9,15,34,37,32,13,10,16,31,50,8,14,7,29)(5,17,21,11,33,40,19,48,22,52,61,41,20,46,35,39)(6,23,26,42,25,45,30,49,28,44,36,47,27,12,24,43)(18,53,56,62,55,63,54,38,58,65,60,64,57,51,59,66);(3,5,8,33,10,22,34,20)(4,11,37,39,16,41,14,48)(6,18,27,55,28,58,25,57)(7,21,31,19,32,61,9,35)(12,38,23,62,45,66,44,64)(13,46,50,52,29,40,15,17)(24,56,36,54,30,60,26,59)(42,53,49,51,47,65,43,63);(3,6)(4,12)(5,18)(7,30)(8,27)(9,36)(10,28)(11,38)(13,47)(14,44)(15,49)(16,45)(17,51)(19,59)(20,57)(21,60)(22,58)(23,37)(24,32)(25,34)(26,31)(29,42)(33,55)(35,54)(39,62)(40,53)(41,66)(43,50)(46,65)(48,64)(52,63)(56,61)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401674);