Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
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 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1168 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,3)(9,10,11,12)(13,14)(15,22)(16,19)(17,20)(18,21)(23,28)(24,25)(26,27);(5,6)(7,8)(13,15,22,14,18,20,17,21)(16,24,28,27,26,23,25,19);(1,2,4,3)(6,8)(13,16,18,26)(14,19,21,27)(15,23,20,24)(17,25,22,28);(1,3,4,2)(14,20)(15,21)(17,22)(19,24)(23,27)(25,28)Polycyclic presentation:
2^9|a^2=h,b^2=ei,c^2=fi,d^2=f,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, 292432);