Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1242 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,4)(10,13)(11,15)(17,18,23,28,22,26,21,27)(19,29,30,32,20,24,31,25);(5,6)(7,8)(9,10,12,13)(11,16,15,14)(17,19)(18,24)(20,22)(21,31)(23,30)(25,28)(26,29)(27,32);(2,4)(6,8)(17,20,22,19)(18,25,26,32)(21,31,23,30)(24,27,29,28);(2,4)(9,11,12,15)(10,14,13,16)(17,21)(19,30)(20,31)(22,23)(24,29)(27,28)Polycyclic presentation:
2^9|a^2=h,b^2=ei,c^2=i,d^2=ei,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,f^c=fi,h^c=hi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290799);