Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| 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 | |
1938 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,4,8,6)(11,15)(14,16)(17,18,21,20)(19,22,23,24)(25,26)(27,30)(28,32)(29,31);(1,3)(2,6)(4,7)(5,8)(19,23)(22,24)(25,27,29,32)(26,30,31,28);(9,10,13,12)(11,14,15,16)(17,19)(18,22)(20,24)(21,23)(25,27)(26,28)(29,32)(30,31);(1,4,5,6)(2,3,7,8)(9,11)(10,14)(12,16)(13,15);(1,3)(2,4)(5,8)(6,7)(9,12,13,10)(11,16,15,14)(17,20,21,18)(19,24,23,22)(25,28)(26,27)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,c^2=fi,d^2=f,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1293770);