Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#25194 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3013 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3086 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);(7,8,13,17,14,19,12,18)(9,20,22,15,10,24,23,31)(11,26,27,16,29,33,28,34)(21,30,37,38,25,32,36,35);(3,4)(5,6)(7,9,13,22,14,10,12,23)(8,15,17,24,19,31,18,20)(11,25,27,36,29,21,28,37)(16,32,33,35,34,30,26,38);(4,6)(7,10,14,9)(8,15,19,31)(11,25,29,21)(12,23,13,22)(16,30,34,32)(17,20,18,24)(26,38,33,35)(27,37,28,36);(7,11)(8,16)(9,21)(10,25)(12,28)(13,27)(14,29)(15,30)(17,33)(18,26)(19,34)(20,35)(22,37)(23,36)(24,38)(31,32)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346219);