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 | |
2114 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)(4,8)(5,7)(9,10,12,14)(11,15,16,13)(19,23)(22,24)(25,26,29,31)(27,30,32,28);(1,2,5,7)(3,8,4,6)(17,18,21,20)(19,22,23,24)(25,27,29,32)(26,28,31,30);(1,3,5,4)(2,6,7,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,22)(20,24)(21,23)(25,28,29,30)(26,27,31,32);(1,4,5,3)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20,21,18)(19,24,23,22)(25,26)(27,28)(29,31)(30,32);(1,2)(3,6)(4,8)(5,7)(9,10)(11,13)(12,14)(15,16)(17,19)(18,22)(20,24)(21,23)(25,27)(26,30)(28,31)(29,32)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,c^2=f,d^2=gh,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, 1359233);