Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393909 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1444 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,4)(6,7)(9,10,12,14)(11,13,15,16)(17,18)(19,23)(20,24)(21,25)(22,27)(26,30)(28,31)(29,32);(5,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19,22,20)(18,23,27,24)(21,28,30,29)(25,31,26,32);(2,4)(11,15)(13,16)(17,20,22,19)(18,24,27,23)(21,29,30,28)(25,32,26,31);(5,7,8,6)(18,25)(21,30)(23,31)(24,32)(26,27)(28,29)Polycyclic presentation:
2^9|a^2=g,b^2=ei,c^2=i,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 309010);