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 | #2541 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1565 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)(17,18,22,23)(19,27,30,24)(20,25,31,29)(21,26,28,32);(1,3,5,4)(2,6,8,7)(18,23)(19,24)(20,31)(21,26)(27,30)(28,32);(10,13)(14,16)(17,19,22,30)(18,24,23,27)(20,26,31,32)(21,29,28,25);(1,3,5,4)(2,7,8,6)(9,11,12,15)(10,14,13,16)(17,20,22,31)(18,25,23,29)(19,28,30,21)(24,26,27,32);(1,4,5,3)(2,6,8,7)(17,21)(18,26)(19,29)(20,27)(22,28)(23,32)(24,31)(25,30)Polycyclic presentation:
2^9|a^2=i,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, 6496131);