Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4452694 |
| automorphism | Aut(G) order 16777216 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C26 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4018 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,3,4)(5,6,7,8)(11,15)(14,16)(17,18)(19,23)(20,22)(21,28)(24,27)(25,29)(26,31)(30,32);(2,4)(10,13)(14,16)(17,19,22,27)(18,23,20,24)(21,26,29,32)(25,30,28,31);(2,4)(9,10)(11,14)(12,13)(15,16)(17,20,22,18)(19,23,27,24)(21,28,29,25)(26,30,32,31);(1,2)(3,4)(5,6)(7,8)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,30)(29,32);(9,11)(10,14)(12,15)(13,16)(17,21)(18,25)(19,26)(20,28)(22,29)(23,30)(24,31)(27,32)Polycyclic presentation:
2^9|a^2=fgi,b^2=fg,c^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4360393);