Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25140 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#25409 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3153 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2688 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);(4,5)(7,8,12,16,11,15,18,28)(9,17,21,34,20,13,27,30)(10,22,25,32,24,33,26,14)(19,35,36,29,23,38,37,31);(3,4,6,5)(7,9,18,27,11,20,12,21)(8,13,28,34,15,17,16,30)(10,23,26,36,24,19,25,37)(14,31,33,38,32,29,22,35);(7,10,11,24)(8,14,15,32)(9,19,20,23)(12,26,18,25)(13,29,17,31)(16,33,28,22)(21,37,27,36)(30,35,34,38);(3,5,6,4)(8,15)(13,17)(14,32)(16,28)(22,33)(29,31)(30,34)(35,38)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351514);