Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433044 |
| 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×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1415 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,7,4,6)(9,10)(11,13)(12,14)(15,16)(17,18,22,25)(19,26,24,21)(20,28,30,27)(23,31,32,29);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19,21,18,22,24,26,25)(20,28,29,32,30,27,31,23);(1,3,5,4)(2,7,8,6)(11,15)(13,16)(17,20,26,31,22,30,21,29)(18,23,19,27,25,32,24,28);(1,4,5,3)(2,6,8,7)(10,14)(13,16)(17,21,22,26)(18,24,25,19)(20,29,30,31)(23,28,32,27);(17,21,22,26)(18,19,25,24)(20,29,30,31)(23,27,32,28)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=fi,d^2=fg,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,e^b=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6341593);