Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| 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 | C2×#532 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1584 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)(3,8)(4,7)(5,6)(9,10)(11,15)(12,13)(14,16)(17,18,22,27,21,26,28,31)(19,25,30,32,29,23,20,24);(2,6)(3,4)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,27)(25,26)(31,32);(1,3,5,4)(2,7,6,8)(10,13)(14,15)(17,20,21,30)(18,24,26,32)(19,28,29,22)(23,27,25,31);(1,4,5,3)(2,8,6,7)(9,11,12,16)(10,14,13,15)(17,19)(18,25)(20,28)(21,29)(22,30)(23,26)(24,31)(27,32);(2,6)(7,8)(18,26)(19,29)(20,30)(27,31)Polycyclic presentation:
2^9|a^2=h,c^2=f,d^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dfi,e^b=ei,e^c=ei,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6316317);