Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500416 |
| 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×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1546 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,8,4,6)(17,18,22,25)(19,26,29,24)(20,23,31,21)(27,28,30,32);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,30)(21,32)(22,29)(23,28)(25,26)(27,31);(9,11)(10,13)(12,16)(14,15)(17,20,23,25,22,31,21,18)(19,27,28,24,29,30,32,26);(11,16)(13,15)(17,21,22,23)(18,20,25,31)(19,28,29,32)(24,27,26,30);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(19,29)(24,26)(27,30)(28,32)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=g,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6415784);