Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15263 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2327 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,7,8,6)(10,11)(12,16)(17,18,22,23)(19,24,20,21)(25,26)(27,28)(29,32)(30,31);(1,2)(3,6)(4,5)(7,8)(10,11)(14,15)(18,23)(19,20)(25,27)(26,29)(28,30)(31,32);(2,5)(6,7)(9,10,13,11)(12,14,16,15)(17,19,22,20)(18,24,23,21)(25,28,31,29)(26,32,30,27);(1,3,4,8)(2,6,5,7)(9,11,13,10)(12,15,16,14)(17,20,22,19)(18,24,23,21)(25,29,31,28)(26,32,30,27);(1,2)(3,6)(4,5)(7,8)(9,12,13,16)(10,14,11,15)(17,21,22,24)(18,19,23,20)(25,30,31,26)(27,28,32,29)Polycyclic presentation:
2^9|a^2=fhi,c^2=fghi,d^2=h,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2456238);