Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1572 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)(11,15)(12,14)(13,16)(17,18,21,25)(19,26,29,23)(20,31,22,24)(27,28,32,30);(2,6)(7,8)(11,16)(13,15)(17,19,21,29)(18,23,25,26)(20,30,22,28)(24,27,31,32);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,18)(19,26)(20,31)(21,25)(22,24)(23,29)(27,30)(28,32);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,18)(19,27)(20,24)(21,25)(22,31)(23,28)(26,30)(29,32);(1,4,5,3)(2,7,6,8)(17,20,21,22)(18,24,25,31)(19,28,29,30)(23,32,26,27)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fg,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7442963);