Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60454 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29×C4 |
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 | C22×#575 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5214 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)(5,6,7,8)(10,14)(11,17)(12,19)(15,22)(16,21)(18,23);(1,2,3,4)(6,8)(9,10,12,14,13,16,19,21)(11,15,17,20,18,22,23,24);(9,11,12,17,13,18,19,23)(10,15,14,20,16,22,21,24);(11,18)(15,22)(17,23)(20,24);(10,16)(14,21)(15,22)(20,24)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=g,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,e^b=ei,d^c=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72249555, #72249569, #72249594, … |
| maximal normal subgroup of | #72249555, #72249569, #72249594, … |
| maximal quotient of | #72264943, #72310555, #72316033, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279199);