Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C23×#479 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
7986 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 30 points:
(1,2);(3,4);(5,6);(7,8,10,12)(9,13,14,11)(15,16,18,20)(17,23,24,30)(19,27,26,22)(21,28,29,25);(7,9,10,14)(8,11,12,13)(16,20)(21,29)(22,27)(23,30);(15,17,18,24)(16,21,20,29)(19,28,26,25)(22,30,27,23)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482215);