Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| 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 | |
1502 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)(11,14)(12,13)(15,16)(17,18,23,28,22,27,30,31)(19,26,29,32,21,24,20,25);(1,2)(3,6)(4,5)(7,8)(17,19)(18,24)(20,30)(21,22)(23,29)(25,28)(26,27)(31,32);(2,5)(6,8)(10,13)(14,16)(17,20)(18,25)(19,23)(21,30)(22,29)(24,31)(26,28)(27,32);(1,3,4,7)(2,6,5,8)(17,21,22,19)(18,26,27,24)(20,30,29,23)(25,28,32,31);(9,11,12,15)(10,14,13,16)(19,21)(20,29)(24,26)(25,32)Polycyclic presentation:
2^9|a^2=h,d^2=f,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=cf,d^b=di,e^b=ei,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, 6282648);