Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61058 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3010 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3296 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,6,8,7)(9,10)(11,13)(12,22)(14,19)(15,18)(16,20)(17,21)(23,24);(5,7,8,6)(9,11,16,12)(10,17,20,24)(13,19,22,18)(14,21,15,23);(2,4)(9,12,16,11)(10,13,20,22)(14,21,15,23)(17,19,24,18);(9,13)(10,11)(12,20)(14,24)(15,17)(16,22)(18,21)(19,23);(9,14,16,15)(10,18,20,19)(11,21,12,23)(13,24,22,17)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,d^b=df,f^b=fi,e^c=ei,f^d=fi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6288631);