Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#27305 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1832 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4,6,7)(5,8,9,10)(11,12)(13,16)(14,18)(15,17)(19,20)(21,27)(22,25)(23,26)(24,29)(28,32)(30,34)(31,33);(4,7)(8,10)(11,13,15,14)(12,16,17,18)(19,21,22,29)(20,24,25,27)(23,28,31,30)(26,34,33,32);(3,5,6,9)(4,8,7,10)(12,17)(16,18)(20,25)(21,28)(23,31)(24,32)(27,34)(29,30);(4,7)(8,10)(11,14,15,13)(12,18,17,16)(20,25)(24,27)(26,33)(32,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6348060);