Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518952 |
| 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 | #2537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1442 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)(9,10,13,12)(11,14,16,15)(17,18,22,26)(19,27,30,31)(20,25,21,24)(23,29,32,28);(1,3,4,8)(2,5,6,7)(9,11)(10,14)(12,15)(13,16)(17,19,21,28,22,30,20,29)(18,23,25,27,26,32,24,31);(2,6)(3,8)(9,12,13,10)(11,15,16,14)(17,20,22,21)(18,24,26,25)(19,28,30,29)(23,27,32,31);(2,6)(5,7)(11,16)(14,15)(17,20,22,21)(18,24,26,25)(19,29,30,28)(23,31,32,27);(2,6)(5,7)(17,19,20,29,22,30,21,28)(18,23,24,31,26,32,25,27)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=hi,d^2=i,e^2=fg,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7514063);