Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5955303 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
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 | |
1722 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,4)(9,10)(11,14)(12,13)(15,16)(18,22)(23,24)(25,26,30,32)(27,29,31,28);(1,2,6,5)(3,8,4,7)(10,13)(14,16)(17,18,21,22)(19,23,20,24)(25,27,30,31)(26,28,32,29);(1,3,6,4)(2,7,5,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,28)(29,31)(30,32);(1,4,6,3)(2,8,5,7)(17,19,21,20)(18,23,22,24)(25,28,30,29)(26,27,32,31);(1,5,6,2)(3,8,4,7)(9,11,12,15)(10,14,13,16)(17,20,21,19)(18,24,22,23)(25,29,30,28)(26,31,32,27)Polycyclic presentation:
2^9|a^2=fh,b^2=g,c^2=i,d^2=g,e^2=f,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, 5863190);