Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C43 · Schur multiplier (H2) = C22×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C4 | C2×C43 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1642 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,4,3)(13,14,17,19,18,22,16,21)(15,20,25,26,23,27,24,28);(5,6,7,8)(14,19)(16,17)(20,26)(21,22)(24,25)(27,28);(9,10,11,12)(13,15)(14,20)(16,24)(17,25)(18,23)(19,26)(21,28)(22,27);(1,3,4,2)(15,23)(20,27)(24,25)(26,28)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,d^2=ef,e^2=i,f^2=i,b^a=be,e^b=ei,f^b=fi,d^c=di176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260419);