Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9495 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#9518 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1426 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4,7,12,8,11,6,10)(5,9,13,16,14,17,15,18)(19,20)(21,24,28,32)(22,29,30,25)(23,26)(27,31)(33,34);(3,5,6,13,8,14,7,15)(4,9,12,18,11,17,10,16)(19,21,23,28)(20,24)(22,27,30,33)(25,31)(26,32)(29,34);(3,6,8,7)(4,10,11,12)(5,13,14,15)(9,16,17,18)(19,22,23,30)(20,25,26,29)(21,27,28,33)(24,31,32,34)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 109059);