Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle, prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1690 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,6,7,8)(9,10)(11,14)(12,13)(15,16)(18,23)(19,24)(20,21);(1,2)(3,4)(10,13)(14,16)(17,18,22,24)(19,21,23,20);(2,4)(17,19,22,23)(18,20,24,21);(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=i,d^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,e^c=ei,f^c=fi128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265874);