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×#27551 |
| 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 | |
1880 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);(3,4,7,10)(5,9,6,8)(11,12,14,15)(13,17,18,16)(19,20)(21,24)(22,25)(23,26)(27,34)(28,32)(29,31)(30,33);(3,5,7,6)(4,8,10,9)(12,15)(16,17)(19,21,22,28)(20,24,25,32)(23,27,30,29)(26,34,33,31);(3,5,7,6)(4,9,10,8)(11,13,14,18)(12,16,15,17)(20,25)(21,27)(23,30)(24,31)(28,29)(32,34);(3,6,7,5)(4,9,10,8)(12,15)(16,17)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6348241);