Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1394 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,7,8)(9,10,12,11)(13,14)(15,21)(16,23)(17,22)(18,19)(20,27)(24,25)(26,28);(1,2)(3,4)(13,15,21,14,18,22,17,19)(16,24,27,28,26,20,25,23);(2,4)(6,8)(9,11,12,10)(14,19)(15,22)(20,24)(23,28);(6,8)(13,16,18,26)(14,20,19,24)(15,23,22,28)(17,27,21,25)Polycyclic presentation:
2^9|a^2=h,b^2=ei,c^2=fhi,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=dei,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 296305);