Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573511 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2401 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1383 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)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,18,22,23)(19,20,24,26)(21,25,30,31)(27,28,32,29);(2,6)(7,8)(9,11,12,15)(10,13,14,16)(17,19,22,24)(18,20,23,26)(21,29,30,28)(25,27,31,32);(1,3,5,4)(2,7,6,8)(10,14)(13,16)(18,23)(19,26)(20,24)(21,30)(27,28)(29,32);(1,4,5,3)(2,8,6,7)(17,20)(18,24)(19,23)(21,27)(22,26)(25,28)(29,31)(30,32);(17,21,22,30)(18,25,23,31)(19,27,24,32)(20,28,26,29)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=f,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dh,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6488029);