Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#28727 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 30 points:
(1,2);(7,8)(9,13)(10,12)(11,14)(15,16)(17,23)(18,20)(19,27)(21,26)(22,24)(25,29)(28,30);(3,4,5,6)(7,9,11,10)(8,12,14,13)(16,20)(21,28)(22,27)(23,29);(4,6)(7,9,11,10)(8,13,14,12)(15,17,24,30,18,25,19,26)(16,21,27,29,20,28,22,23);(7,10,11,9)(8,13,14,12)(16,20)(17,25)(22,27)(26,30)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6373716);