Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C42 |
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 | |
2451 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)(3,5)(4,6)(7,8)(9,10)(11,16)(12,13)(14,15)(17,18)(19,22)(20,21)(23,24)(25,26,30,29)(27,32,28,31);(2,3)(4,8)(9,11)(10,13)(12,14)(15,16)(18,20)(19,23)(25,27,30,28)(26,31,29,32);(2,3)(6,7)(10,15)(11,12)(17,18,21,20)(19,24,23,22)(25,26,30,29)(27,31,28,32);(1,3,5,2)(4,7,8,6)(9,12,14,11)(10,16,15,13)(17,19,21,23)(18,22,20,24)(25,28,30,27)(26,31,29,32);(1,4,5,8)(2,6,3,7)(9,13,14,16)(10,11,15,12)(17,20,21,18)(19,24,23,22)(25,29,30,26)(27,31,28,32)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=fi,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4389851);