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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1405 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)(17,18,23,27)(19,26,29,32)(20,25,22,31)(21,30,28,24);(1,3,4,8)(2,5,6,7)(9,10)(11,13)(12,14)(15,16)(17,19,20,21,23,29,22,28)(18,24,31,32,27,30,25,26);(2,6)(3,8)(9,11,12,15)(10,13,14,16);(2,6)(5,7)(10,14)(13,16)(17,20,23,22)(18,25,27,31)(19,28,29,21)(24,32,30,26);(2,6)(5,7)(17,21,20,29,23,28,22,19)(18,26,31,24,27,32,25,30)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=hi,d^2=i,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,e^b=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7507388);