Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2238 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#13140 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #481 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | #1926 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
780 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4,9,15,7,13,10,25)(5,14,8,23,17,11,18,27)(6,19,20,30,22,34,21,26)(12,24,28,16,29,32,31,33);(3,5,10,18,7,17,9,8)(4,11,25,23,13,14,15,27)(6,16,20,32,22,33,21,24)(12,26,28,19,29,30,31,34);(3,6,9,21,7,22,10,20)(4,12,25,28,13,29,15,31)(5,16,8,24,17,33,18,32)(11,26,23,19,14,30,27,34)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402078);