Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5955303 |
| automorphism | Aut(G) order 8388608 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2634 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 32 points:
(6,8)(9,10,12,14)(11,13,15,16)(17,18,22,26)(19,25,21,23)(20,28,30,24)(27,31,29,32);(2,4)(5,6)(7,8)(17,19,22,21)(18,23,26,25)(20,29,30,27)(24,32,28,31);(1,2,3,4)(9,10)(11,13)(12,14)(15,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,31)(28,30)(29,32);(17,20,22,30)(18,24,26,28)(19,27,21,29)(23,31,25,32);(9,11,12,15)(10,13,14,16)(17,21,22,19)(18,25,26,23)(20,29,30,27)(24,32,28,31)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,d^2=i,e^2=g,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, 5863673);