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 | #15320 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2402 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)(3,7)(4,8)(5,6)(9,10,13,16)(11,15,12,14)(17,18)(19,24)(20,21)(22,23)(25,26,30,29)(27,28,32,31);(2,6)(4,7)(11,12)(14,15)(18,23)(21,24)(25,27,30,32)(26,31,29,28);(9,11,13,12)(10,14,16,15)(17,19,22,20)(18,24,23,21)(25,28)(26,32)(27,29)(30,31);(1,3)(2,7)(4,6)(5,8)(9,12,13,11)(10,15,16,14)(17,20,22,19)(18,24,23,21)(25,28)(26,27)(29,32)(30,31);(1,4,5,7)(2,8,6,3)(9,10)(11,14)(12,15)(13,16)(17,21,22,24)(18,20,23,19)(25,29,30,26)(27,31,32,28)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=ghi,d^2=ghi,e^2=f,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, 2860601);