Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #2407 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1548 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,7)(9,10)(11,13)(12,14)(15,16)(17,18,21,20,22,24,29,28)(19,23,27,25,26,30,32,31);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,25)(21,27)(22,26)(24,30)(28,31)(29,32);(10,14)(11,15)(17,19)(18,23)(20,25)(21,27)(22,26)(24,30)(28,31)(29,32);(2,5)(6,8)(10,14)(13,16)(17,20,21,24,22,28,29,18)(19,25,27,30,26,31,32,23);(1,3,4,7)(2,6,5,8)(19,26)(23,30)(25,31)(27,32)Polycyclic presentation:
2^9|a^2=h,d^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=df,e^b=ei,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72284940, #72285096, #72285118, … |
| maximal normal subgroup of | #72284940, #72285096, #72285118, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6328078);