Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
| direct product | C4×#2822 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2214 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,3,4);(10,11)(14,15)(17,18,21,24)(19,25,27,22)(20,23,29,32)(26,31,28,30);(5,6)(7,8)(9,10,13,11)(12,14,16,15)(17,19,21,27)(18,22,24,25)(20,28,29,26)(23,31,32,30);(6,8)(9,11,13,10)(12,14,16,15)(20,29)(23,32)(26,28)(30,31);(9,12,13,16)(10,14,11,15)(17,20,21,29)(18,23,24,32)(19,26,27,28)(22,30,25,31)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 451534);