Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2397 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3286 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(5,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,24)(21,23)(25,28)(26,29)(27,30)(31,32);(1,2)(3,4)(10,13)(11,15)(17,19)(18,22)(20,26)(21,25)(23,28)(24,29)(27,31)(30,32);(2,4)(11,15)(14,16)(18,23)(22,28)(24,30)(29,32);(9,11,12,15)(10,14,13,16)(19,25)(22,28)(26,31)(29,32);(17,20,21,27)(18,24,23,30)(19,26,25,31)(22,29,28,32)Polycyclic presentation:
2^9|a^2=i,d^2=f,e^2=g,b^a=bf,c^a=cg,c^b=ch,d^b=dfg,d^c=df104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 451474);