Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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 | #2714 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1388 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,5)(4,6)(7,8)(9,10,13,16)(11,15,12,14)(17,18,23,28,22,27,29,30)(19,24,20,21,26,32,31,25);(1,3,4,7)(2,5,6,8)(9,10)(11,14)(12,15)(13,16)(17,19,22,26)(18,24,27,32)(20,29,31,23)(21,30,25,28);(2,6)(5,8)(9,11,13,12)(10,14,16,15)(17,20)(18,25)(19,29)(21,27)(22,31)(23,26)(24,28)(30,32);(9,11,13,12)(10,15,16,14)(19,26)(20,31)(21,25)(24,32);(9,12,13,11)(10,15,16,14)(17,21,23,32,22,25,29,24)(18,26,28,31,27,19,30,20)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=f,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,e^a=ef,d^b=dfi,d^c=di,e^c=ei,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6309333);