Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454382 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1452 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,4)(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,27)(20,25)(21,24)(22,26)(23,30)(28,31)(29,32);(10,13)(14,16)(17,19,22,29,21,30,20,28)(18,23,26,31,24,27,25,32);(1,2,3,4)(9,11)(10,14)(12,15)(13,16)(18,24)(19,28)(20,22)(23,31)(27,32)(29,30);(6,8)(11,15)(14,16)(17,20,21,22)(18,25,24,26)(19,29,30,28)(23,31,27,32)Polycyclic presentation:
2^9|b^2=g,c^2=fi,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 372119);