Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573485 |
| 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×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#125 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1340 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,7,4,6)(9,10)(11,14)(12,13)(15,16)(17,18,24,23)(19,21,26,32)(20,25,22,27)(28,31,29,30);(1,3,5,4)(2,6,8,7)(17,19,24,26)(18,21,23,32)(20,30,22,31)(25,28,27,29);(10,13)(14,16)(17,20,24,22)(18,25,23,27)(19,28,26,29)(21,31,32,30);(1,3,5,4)(2,7,8,6)(9,11,12,15)(10,14,13,16)(17,21,24,32)(18,26,23,19)(20,28,22,29)(25,31,27,30);(1,4,5,3)(2,6,8,7)(17,22,24,20)(18,27,23,25)(19,29,26,28)(21,30,32,31)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=i,d^2=fgi,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6496419);