Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7734 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×#7805 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #120 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
1001 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(4,8)(5,6)(9,10)(12,15)(13,16)(17,18)(19,20,23,26)(21,24,28,31)(22,25,29,32)(27,33,34,30);(3,4)(5,8)(6,10)(7,9)(11,12,14,15)(13,16,17,18)(19,21,23,28)(20,24,26,31)(22,27,29,34)(25,30,32,33);(3,5,7,6)(4,8,9,10)(11,13)(12,16)(14,17)(15,18)(19,22)(20,25)(21,27)(23,29)(24,30)(26,32)(28,34)(31,33)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 72978);