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 | #34911 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1618 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)(3,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18,24,19,23,28,22,27)(20,29,21,31,30,25,32,26);(1,3)(2,5)(4,8)(6,7)(17,19,24,28,23,27,22,18)(20,26,21,29,30,31,32,25);(3,8)(5,7)(10,13)(14,16)(17,20)(18,25)(19,26)(21,24)(22,32)(23,30)(27,31)(28,29);(2,6)(5,7)(17,21)(18,26)(19,29)(20,22)(23,32)(24,30)(25,27)(28,31);(9,11,12,15)(10,14,13,16)(17,22,23,24)(18,27,28,19)(20,21,30,32)(25,26,29,31)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=cfi,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, 6284350);