Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C2×D4×#41 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5346 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(6,8)(9,10,15,16,14,11,12,18)(13,17,22,23,21,19,20,24);(1,2,3,4)(9,11,15,18,14,10,12,16)(13,19,22,24,21,17,20,23);(5,6,7,8)(9,12,14,15)(10,16,11,18)(13,20,21,22)(17,23,19,24);(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24);(13,21)(17,19)(20,22)(23,24)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=g,h^2=i,b^a=bf,c^a=cg,c^b=ci,e^d=ei128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279253);