Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24986 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#28250 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | #2537 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2466 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4)(5,15)(6,20)(7,10)(8,13)(9,14)(11,18)(12,22)(16,25)(17,26)(19,27)(21,24)(23,28)(29,30)(31,33)(32,34);(4,10,14,13)(5,16,19,18)(6,21)(11,25)(12,20,28,24)(15,27)(17,31,32,30)(22,23);(3,5)(4,11)(6,17)(7,16)(8,18)(9,19)(10,15)(12,26)(13,27)(14,25)(20,33)(21,32)(22,30)(23,31)(24,29)(28,34);(3,6,7,22,9,21,8,23)(4,12,13,20,14,28,10,24)(5,17,16,30,19,32,18,31)(11,26,27,33,25,34,15,29)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6372713);