Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5955303 |
| automorphism | Aut(G) order 16777216 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3358 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)(10,13)(14,15)(19,25)(20,28)(22,26)(23,30);(2,4)(9,10)(11,15)(12,13)(14,16)(17,18,21,24)(19,26,25,22)(20,23,28,30)(27,32,31,29);(1,2)(3,4)(5,6)(7,8)(9,11,12,16)(10,14,13,15)(17,18)(19,22)(20,23)(21,24)(25,26)(27,29)(28,30)(31,32);(17,19,21,25)(18,22,24,26)(20,27,28,31)(23,29,30,32);(17,20)(18,23)(19,27)(21,28)(22,29)(24,30)(25,31)(26,32)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=f,d^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5863627);