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 | |
1977 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,3)(6,7)(9,10)(11,15)(12,14)(13,16)(17,18,21,20)(19,23,24,22)(26,28)(27,29);(1,2,5,3)(4,7,8,6)(9,10,12,14)(11,13,16,15)(17,19)(18,22)(20,23)(21,24)(25,26)(27,31)(28,30)(29,32);(2,3)(6,7)(11,16)(13,15)(17,19,21,24)(18,23,20,22)(25,27,30,29)(26,31,28,32);(1,3,5,2)(4,7,8,6)(9,11,12,16)(10,13,14,15)(17,19)(18,23)(20,22)(21,24)(25,28,30,26)(27,31,29,32);(1,4,5,8)(2,6,3,7)(9,10)(11,13)(12,14)(15,16)(17,20,21,18)(19,22,24,23)(25,29,30,27)(26,31,28,32)Polycyclic presentation:
2^9|a^2=fi,b^2=gi,c^2=fgi,d^2=i,e^2=gh,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, 1786495);