Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9443 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#9456 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | C2×C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1162 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)(19,20,25,28,24,21,23,27)(22,26,32,34,31,29,30,33);(3,5,6,13,8,14,7,15)(4,9,12,18,11,17,10,16)(19,21,24,20)(22,29,31,26)(23,27,25,28)(30,33,32,34);(3,6,8,7)(4,10,11,12)(5,13,14,15)(9,16,17,18)(19,22,24,31)(20,26,21,29)(23,32,25,30)(27,34,28,33)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 108389);