Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 | C2×#525 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1651 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,15)(12,16)(13,14)(17,18,22,25,21,24,30,31)(19,26,28,32,27,20,23,29);(1,3)(2,5)(4,8)(6,7)(17,19,22,28,21,27,30,23)(18,20,25,29,24,26,31,32);(10,14)(15,16)(17,20)(18,23)(19,25)(21,26)(22,29)(24,28)(27,31)(30,32);(3,8)(5,7)(9,11,13,12)(10,15,14,16);(2,6)(5,7)(9,12,13,11)(10,16,14,15)(18,24)(20,26)(25,31)(29,32)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=g,e^2=g,h^2=i,b^a=bf,c^a=cg,e^a=ef,d^b=dfi,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6299065);