Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D4×D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#33853 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
2042 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(3,4)(5,6)(8,13)(9,18)(10,19)(11,20)(14,25)(15,21)(16,31)(17,29)(23,24)(26,36)(27,28)(30,35)(32,38)(34,37);(7,8,12,16)(9,14,18,17)(10,21,23,37)(11,25,27,29)(13,28,31,20)(15,33,34,22)(19,30,24,38)(26,35,36,32);(4,6)(7,9,12,18)(8,14,16,17)(10,19,23,24)(11,20,27,28)(13,29,31,25)(15,32,34,35)(21,30,37,38)(22,36,33,26);(7,10,12,23)(8,15,16,34)(9,19,18,24)(11,26,27,36)(13,30,31,38)(14,32,17,35)(20,22,28,33)(21,29,37,25)110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530798);