Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1737 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)(19,26)(20,23)(21,25)(22,29)(24,28)(27,31)(30,32);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(19,27)(22,30)(26,31)(29,32);(1,3,5,4)(2,7,8,6)(10,14)(11,15)(17,19,28,32,21,27,20,29)(18,22,23,31,25,30,24,26);(1,4,5,3)(2,6,8,7)(11,15)(13,16);(17,20,21,28)(18,23,25,24)(19,29,27,32)(22,31,30,26)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=fgh,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dhi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6367781);