Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#27826 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1978 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);(3,4,7,10)(5,9,6,8)(11,12)(13,15)(14,16)(17,18)(19,20,24,26,22,27,23,28)(21,29,30,33,31,25,32,34);(3,5,7,6)(4,8,10,9)(11,13,14,17)(12,15,16,18)(19,21,24,30,22,31,23,32)(20,25,26,34,27,29,28,33);(12,16)(15,18)(20,26)(21,30)(23,24)(25,29)(27,28)(31,32);(3,6,7,5)(4,9,10,8)(20,27)(21,31)(26,28)(30,32)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6365525);