Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #2733 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1506 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,24,28,23,27,21,26)(19,29,30,32,20,25,31,22);(1,2)(3,7)(4,8)(5,6)(9,11,12,15)(10,13,14,16)(17,19)(18,25)(20,23)(21,31)(22,28)(24,30)(26,32)(27,29);(1,3,5,4)(2,7,6,8)(10,14)(13,16)(17,20,23,19)(18,25,27,29)(21,30,24,31)(22,26,32,28);(1,4,5,3)(2,7,6,8)(17,21,23,24)(18,26,27,28)(19,30,20,31)(22,29,32,25);(17,22,24,29,23,32,21,25)(18,19,28,30,27,20,26,31)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=f,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,c^b=ci,d^b=df,d^c=di,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6292850);