Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18132 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#18155 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2918 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4,7,9)(5,11,13,18)(6,14,16,10)(8,15,17,12)(24,30)(31,34);(3,5,7,13)(4,8,9,17)(6,15,16,12)(10,18,14,11)(19,20)(21,24)(22,25)(23,26)(27,31)(28,30)(29,32)(33,34);(19,21)(20,24)(22,27)(23,28)(25,31)(26,30)(29,33)(32,34);(4,9)(8,17)(10,14)(11,18)(19,22,23,29)(20,25,26,32)(21,27,28,33)(24,31,30,34)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6253595);