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 | |
2029 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,5)(3,4,8,7)(9,10,12,13)(11,15,16,14)(17,18,21,20)(19,22,23,24)(27,29)(31,32);(1,3,6,8)(2,7,5,4)(10,13)(14,15)(18,20)(22,24)(25,26,30,28)(27,31,29,32);(9,11)(10,14)(12,16)(13,15)(17,18,21,20)(19,22,23,24)(25,27,30,29)(26,31,28,32);(1,4)(2,3)(5,8)(6,7)(9,11)(10,15)(12,16)(13,14)(17,19)(18,22)(20,24)(21,23)(25,28,30,26)(27,32,29,31);(1,5,6,2)(3,7,8,4)(9,10)(11,14)(12,13)(15,16)(17,20,21,18)(19,22,23,24)(25,29,30,27)(26,32,28,31)Polycyclic presentation:
2^9|a^2=f,b^2=fgh,c^2=fgi,d^2=fghi,e^2=fg,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, 1406771);