Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C25×C42 |
| direct product | C2×#23351 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C24 | C23×C4 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1946 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(4,7)(5,9)(11,12)(13,17)(14,16)(15,18)(19,20,24,28,23,25,22,27)(21,29,31,34,30,26,32,33);(3,4)(5,10)(6,7)(8,9)(13,18)(15,17)(20,25)(21,30)(27,28)(31,32);(4,7)(8,10)(11,13,14,18)(12,15,16,17)(19,21,23,30)(20,26,25,29)(22,32,24,31)(27,34,28,33);(3,5,6,9)(4,8,7,10)(12,16)(15,17)(19,22,23,24)(20,27,25,28)(21,31,30,32)(26,33,29,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6314806);