Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| 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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2025 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)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(19,20)(22,23)(26,30)(28,31);(1,3)(2,5)(4,8)(6,7)(9,10,12,14)(11,13,16,15)(17,18,21,24)(19,22,20,23)(26,30)(27,32);(3,8)(5,6)(11,16)(13,15)(17,19,21,20)(18,22,24,23)(25,26)(27,31)(28,32)(29,30);(1,3)(2,6)(4,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,18)(19,22)(20,23)(21,24)(25,27,29,32)(26,31,30,28);(1,2)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,20,21,19)(18,23,24,22)(25,28,29,31)(26,32,30,27)Polycyclic presentation:
2^9|b^2=fgh,c^2=fghi,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,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1428272);