Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1418353 |
| automorphism | ? |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2000 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,7)(3,6,8,5)(10,13)(11,15)(19,21)(20,23)(25,26)(27,31)(28,30)(29,32);(1,3,4,8)(2,5,7,6)(9,10)(11,16)(12,13)(14,15)(17,18,22,24)(19,23,21,20)(25,27,30,29)(26,31,28,32);(9,10)(11,14)(12,13)(15,16)(17,19,22,21)(18,23,24,20)(25,28,30,26)(27,31,29,32);(1,2)(3,5)(4,7)(6,8)(9,10)(11,16)(12,13)(14,15)(17,20)(18,19)(21,24)(22,23);(1,3)(2,6)(4,8)(5,7)(9,11,12,15)(10,14,13,16)(17,21,22,19)(18,20,24,23)(25,29,30,27)(26,31,28,32)Polycyclic presentation:
2^9|a^2=ghi,b^2=fg,c^2=fhi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1389665);