Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1643 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,14)(12,13)(15,16)(17,18,22,23)(19,27,29,24)(20,25,28,30)(21,26,31,32);(1,3,4,8)(2,5,6,7)(17,19,22,29)(18,24,23,27)(20,26,28,32)(21,30,31,25);(10,13)(14,16)(17,20,22,28)(18,25,23,30)(19,21,29,31)(24,32,27,26);(9,11,12,15)(10,14,13,16)(17,21)(18,26)(19,28)(20,29)(22,31)(23,32)(24,25)(27,30);(17,18,22,23)(19,24,29,27)(20,30,28,25)(21,26,31,32)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=i,d^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6356522);