Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2190 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#12944 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | #1862 |
| 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 | |
1046 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);(3,4,8,14,10,15,9,28)(5,16,20,32,21,12,25,31)(6,22,19,30,23,29,18,11)(7,24,17,33,27,34,26,13);(3,5,8,20,10,21,9,25)(4,11,14,22,15,30,28,29)(6,17,19,27,23,26,18,7)(12,13,31,24,16,33,32,34);(3,6)(4,12)(5,17)(7,25)(8,19)(9,18)(10,23)(11,13)(14,31)(15,16)(20,27)(21,26)(22,24)(28,32)(29,34)(30,33)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401505);