Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417589 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1772 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,4)(10,13)(14,15)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32);(5,6)(7,8)(9,10,12,13)(11,15,16,14)(17,19,22,21)(18,23,26,25)(20,27,29,28)(24,30,32,31);(2,4)(6,8)(17,20,22,29)(18,24,26,32)(19,27,21,28)(23,30,25,31);(9,11,12,16)(10,14,13,15)(18,25)(19,21)(23,26)(24,31)(27,28)(30,32)Polycyclic presentation:
2^9|b^2=ei,c^2=i,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=dei,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325746);