Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C4 |
| direct product | C4×#2809 |
| 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 | |
2210 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4);(5,6,9,11)(7,8,12,10)(13,14)(15,18)(16,19)(17,20)(21,22)(23,27)(24,26)(25,28);(5,7,9,12)(6,10,11,8)(13,15,17,16)(14,18,20,19)(21,23)(22,25)(24,28)(26,27);(13,15,17,16)(14,19,20,18)(22,26)(23,28);(5,8,9,10)(6,7,11,12)(13,16,17,15)(14,18,20,19)(23,28)(25,27)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 428448);