Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32941 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1542 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6)(7,8)(9,10,14,17,15,12,13,23)(11,18,21,19,22,24,20,16);(9,11,13,20,15,22,14,21)(10,16,17,18,12,19,23,24);(1,2,3,4);(6,8)(9,12)(10,15)(11,19)(13,23)(14,17)(16,22)(18,21)(20,24)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^d=gi,h^d=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299666);