Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13392 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
| direct product | C2×#13513 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #482 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | #2085 |
| prime radical | C24 | #137 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
528 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);(3,4,7,25,8,14,10,13)(5,16,18,11,20,36,32,37)(6,12,22,41,24,42,33,39)(9,28,26,43,27,15,31,59)(17,48,49,64,50,46,51,34)(19,44,29,63,53,47,60,35)(21,54,55,38,56,66,52,65)(23,57,30,62,58,40,61,45);(3,5,7,18,8,20,10,32)(4,11,13,16,14,37,25,36)(6,21,22,55,24,56,33,52)(9,29,26,53,27,60,31,19)(12,38,39,54,42,65,41,66)(15,44,43,35,28,47,59,63)(17,30,49,58,50,61,51,23)(34,62,46,57,64,45,48,40);(3,6,7,22,8,24,10,33)(4,12,13,39,14,42,25,41)(5,17,18,49,20,50,32,51)(9,30,26,58,27,61,31,23)(11,34,16,46,37,64,36,48)(15,45,43,40,28,62,59,57)(19,52,29,21,53,55,60,56)(35,54,47,65,63,66,44,38)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406613);