Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 | C2×#466 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2401 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1324 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,6,8,7)(9,10,14,15)(11,12,16,13)(17,18)(19,21)(20,29)(22,28)(23,26)(24,27)(25,31)(30,32);(1,3)(2,6)(4,7)(5,8)(9,11)(10,12)(13,15)(14,16)(17,19,21,18,24,26,23,27)(20,30,32,29,31,22,28,25);(1,4,5,2)(3,7,8,6)(10,15)(11,16)(17,20,24,31)(18,25,27,29)(19,22,26,30)(21,32,23,28);(3,8)(6,7)(9,12,14,13)(10,11,15,16)(17,21,24,23)(18,26,27,19)(20,28,31,32)(22,29,30,25);(9,13,14,12)(10,11,15,16)(17,22,23,29,24,30,21,25)(18,20,19,28,27,31,26,32)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=gh,d^2=g,e^2=fgi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7515298);