Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518992 |
| 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×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1463 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,7,4,8)(9,10,12,14)(11,13,15,16)(17,18,24,19)(20,29,31,26)(21,25,32,28)(22,27,23,30);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,24,18)(20,26,31,29)(21,28,32,25)(22,30,23,27);(1,3,5,4)(2,7,6,8)(17,20,22,32,24,31,23,21)(18,25,30,26,19,28,27,29);(1,3,5,4)(2,7,6,8)(11,15)(13,16)(17,21,23,31,24,32,22,20)(18,26,27,25,19,29,30,28);(1,4,5,3)(2,7,6,8)(17,22,24,23)(18,27,19,30)(20,21,31,32)(25,26,28,29)Polycyclic presentation:
2^9|a^2=fgh,b^2=i,c^2=fgi,d^2=fg,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,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, 7501486);