Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32884 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1446 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(6,8)(9,10,15,17,14,13,12,18)(11,19,22,16,23,24,20,21);(2,4)(9,11,15,22,14,23,12,20)(10,16,17,24,13,21,18,19);(5,6,7,8)(9,12,14,15)(10,17,13,18)(11,20,23,22)(16,24,21,19);(9,13)(10,14)(11,21)(12,17)(15,18)(16,23)(19,22)(20,24)Polycyclic presentation:
2^9|a^2=h,b^2=h,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^a=gi,e^b=ei,g^b=gi,d^c=di,e^d=ei,g^d=gi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300139);