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×#33844 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C2×#3287 | 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 | |
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)(7,8,13,17)(9,18,19,37)(10,23,15,35)(11,26,29,25)(12,30,28,16)(14,32,33,21)(20,38,34,27)(22,24,31,36);(7,9,13,19)(8,14,17,33)(10,24,15,36)(11,27,29,38)(12,31,28,22)(16,20,30,34)(18,25,37,26)(21,35,32,23);(4,6)(9,19)(14,33)(18,37)(20,34)(21,32)(22,31)(24,36)(27,38);(7,10,13,15)(8,11,17,29)(9,20,19,34)(12,26,28,25)(14,22,33,31)(16,35,30,23)(18,32,37,21)(24,38,36,27)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530788);