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 | C22×D4 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1364 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)(3,8)(4,6)(5,7)(9,10,12,14)(11,15,16,13)(17,18,23,25,22,27,20,29)(19,24,28,32,26,31,30,21);(9,11)(10,13)(12,16)(14,15)(17,19)(18,24)(20,30)(21,29)(22,26)(23,28)(25,32)(27,31);(1,3,5,4)(2,6,7,8)(10,14)(13,15)(17,20,22,23)(18,25,27,29)(19,28,26,30)(21,31,32,24);(11,16)(13,15)(19,26)(21,32)(24,31)(28,30);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,21,20,31,22,32,23,24)(18,26,29,28,27,19,25,30)Polycyclic presentation:
2^9|a^2=h,c^2=fg,e^2=gh,h^2=i,b^a=bf,c^a=cg,e^a=efi,c^b=ci,d^b=dfi,e^b=ei,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6314391);