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×#33851 |
| 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 | |
1984 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)(9,17)(10,22)(11,25)(14,18)(15,34)(16,30)(20,21)(23,27)(24,36)(26,28)(29,35)(32,38)(33,37);(7,8,12,16)(9,18,21,37)(10,23,26,34)(11,27,25,15)(13,28,30,22)(14,31,33,19)(17,29,20,38)(24,35,36,32);(4,6);(7,9,12,21)(8,14,16,33)(10,24,26,36)(11,17,25,20)(13,29,30,38)(15,35,27,32)(18,34,37,23)(19,28,31,22)110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530796);