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 | #52169 | C2 |
| 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)(3,7)(9,10)(11,13)(12,14)(15,16)(17,18,23,19,22,24,21,26)(20,28,31,30,29,25,32,27);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,15)(14,16)(18,24)(19,26)(20,29)(31,32);(10,14)(11,15)(17,19,23,24,22,26,21,18)(20,30,31,25,29,27,32,28);(2,5)(6,8)(10,14)(13,16)(17,20)(18,25)(19,27)(21,32)(22,29)(23,31)(24,28)(26,30);(1,3,4,7)(2,6,5,8)(17,21,22,23)(18,19,24,26)(20,31,29,32)(25,30,28,27)Polycyclic presentation:
2^9|a^2=h,c^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df,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, 6319072);