Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32894 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C212 |
| direct product | C2×#32896 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2846 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×H(7,𝔽2) | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
17016 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4,9,15)(5,16,19,29)(6,12,21,28)(7,14,8,13)(10,22,26,18)(11,23,27,20)(17,24,34,31)(25,33,30,32);(3,5)(4,10)(6,17)(7,22)(8,18)(9,19)(11,24)(12,25)(13,29)(14,16)(15,26)(20,33)(21,34)(23,32)(27,31)(28,30);(3,6)(4,11)(5,17)(7,20)(8,23)(9,21)(10,24)(12,13)(14,28)(15,27)(16,30)(18,32)(19,34)(22,33)(25,29)(26,31);(3,6)(4,12)(5,17)(7,23)(8,20)(9,21)(10,25)(11,13)(14,27)(15,28)(16,31)(18,33)(19,34)(22,32)(24,29)(26,30)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536075);