Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C22×#2101 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2718 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)(7,12,8,10)(13,14)(15,17)(16,18)(19,20)(21,22,24,25)(23,26,28,27);(5,7,9,8)(6,10,11,12)(13,15,16,19)(14,17,18,20)(21,23,24,28)(22,26,25,27);(5,8,9,7)(6,10,11,12)(14,18)(17,20)(22,25)(23,27)(26,28)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339627);