Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| 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, cosocle | C24 | C25 |
| prime radical | #15310 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2659 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,15)(14,16)(17,18)(19,23)(20,21)(22,24)(25,26)(27,28)(29,32)(30,31);(2,5)(6,7)(10,11)(12,16)(18,20)(19,24)(25,27,31,32)(26,29,30,28);(9,10,13,11)(12,14,16,15)(17,19)(18,22)(20,23)(21,24)(25,28,31,29)(26,32,30,27);(1,3,4,8)(2,6,5,7)(9,11,13,10)(12,14,16,15)(17,19)(18,23)(20,22)(21,24)(25,29,31,28)(26,32,30,27);(1,2)(3,7)(4,5)(6,8)(9,12,13,16)(10,14,11,15)(17,20,21,18)(19,23,24,22)(25,30,31,26)(27,29,32,28)Polycyclic presentation:
2^9|b^2=gh,c^2=ghi,d^2=fhi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2859825);