Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417581 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
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 | |
1154 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,3)(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,26)(20,22)(21,23)(24,27)(25,28)(29,30)(31,32);(1,3,4,2)(9,11,12,15)(10,13,14,16)(18,22)(19,27)(20,28)(23,25)(26,30)(29,32);(2,3)(10,14)(13,16)(18,23)(22,25)(24,31)(26,30);(6,8)(17,19,28,32,21,29,20,27)(18,24,22,30,23,31,25,26)Polycyclic presentation:
2^9|a^2=g,b^2=fi,d^2=ei,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,h^b=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 340264);