Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#25186 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3069 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(3,4,6,5)(7,8,12,17)(9,18,22,14)(10,23,25,32)(11,13,27,16)(15,26,31,24)(19,28,21,30)(20,35,36,38)(29,34,33,37);(3,5,6,4)(8,13)(9,19)(11,27)(15,23)(16,17)(20,34)(21,22)(24,26)(28,30)(31,32)(35,38)(36,37);(4,5)(7,9,12,22)(8,14,17,18)(10,20,25,36)(11,21,27,19)(13,28,16,30)(15,29,31,33)(23,38,32,35)(24,34,26,37);(7,10)(8,15)(9,20)(11,26)(12,25)(13,23)(14,29)(16,32)(17,31)(18,33)(19,34)(21,37)(22,36)(24,27)(28,38)(30,35)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346045);