Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453966 |
| 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 | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1466 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)(6,7)(9,10,13,12)(11,15,16,14)(19,25)(20,27)(22,29)(24,31)(26,28)(30,32);(5,6,8,7)(9,11,13,16)(10,14,12,15)(17,18)(19,22)(20,24)(21,23)(25,29)(26,30)(27,31)(28,32);(6,7)(10,12)(14,15)(17,19,21,26)(18,22,23,30)(20,28,27,25)(24,32,31,29);(2,4)(5,7,8,6)(9,12,13,10)(11,15,16,14)(18,23)(22,30)(24,31)(29,32)Polycyclic presentation:
2^9|a^2=eg,b^2=e,c^2=i,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=di,d^c=de,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367582);