Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#33836 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C2×#3285 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1958 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)(10,21)(11,24)(14,31)(15,35)(16,30)(18,34)(19,32)(20,27)(22,26)(25,28)(29,33)(36,37);(7,8,12,16)(9,18,20,31)(10,22,25,35)(11,26,24,15)(13,28,30,21)(14,32,34,19)(17,37,27,29)(23,36,38,33);(4,6);(7,9,12,20)(8,14,16,34)(10,23,25,38)(11,27,24,17)(13,29,30,37)(15,36,26,33)(18,35,31,22)(19,28,32,21)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530779);