Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13247 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#13248 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1972 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1042 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)(5,17)(6,24,28,54)(7,30)(8,13)(9,34)(10,16)(11,20)(12,27,41,48)(14,32)(15,31)(18,43)(19,35,51,64)(21,37)(22,44)(23,47)(25,55,57,39)(26,58,60,42)(29,62,59,40)(33,46)(36,52)(38,53)(45,50,65,56)(49,63,61,66);(3,5,18,21,9,22,7,20,10,23,8,33,31,52,32,53)(4,11,14,44,15,37,13,17,16,38,30,36,34,46,43,47)(6,25,27,61,29,51,60,50,28,57,48,49,59,19,26,56)(12,39,54,65,42,64,62,66,41,55,24,45,58,35,40,63);(3,6,8,27,9,29,32,60,10,28,18,48,31,59,7,26)(4,12,14,24,15,42,13,40,16,41,30,54,34,58,43,62)(5,19,21,50,22,25,20,49,23,51,33,56,52,57,53,61)(11,35,36,63,37,39,47,65,38,64,44,66,46,55,17,45)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402432);