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 | C2×#121 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1549 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)(6,8)(9,10)(11,14)(12,13)(15,16)(17,18,21,26,22,27,30,32)(19,23,29,31,28,25,20,24);(1,2)(3,6)(4,5)(7,8)(17,19)(18,23)(20,30)(21,29)(22,28)(24,32)(25,27)(26,31);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20)(18,24)(19,21)(22,29)(23,26)(25,32)(27,31)(28,30);(2,5)(6,8)(17,19)(18,25)(20,30)(21,29)(22,28)(23,27)(24,26)(31,32);(9,11,12,15)(10,14,13,16)(19,28)(20,29)(23,25)(24,31)Polycyclic presentation:
2^9|a^2=h,c^2=f,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ci,d^b=df,e^b=ei,d^c=di,e^c=ei,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6288255);