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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1668 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,6,5)(3,8,4,7)(9,10)(11,13)(12,14)(15,16)(17,18,21,24)(19,22,20,23)(25,26,30,31)(27,28,35,32)(29,37,38,33)(34,39,40,36);(1,3,6,4)(2,7,5,8)(9,10,12,14)(11,13,15,16)(26,31)(28,32)(29,38)(34,40);(3,4)(7,8)(9,11,12,15)(10,13,14,16)(17,19,21,20)(18,22,24,23)(25,27,30,35)(26,32,31,28)(29,34,38,40)(33,39,37,36);(1,4,6,3)(2,7,5,8)(9,11)(10,13)(12,15)(14,16)(17,18)(19,22)(20,23)(21,24)(25,28,30,32)(26,27,31,35)(29,36,38,39)(33,34,37,40);(1,5,6,2)(3,7,4,8)(9,10)(11,13)(12,14)(15,16)(17,20,21,19)(18,23,24,22)(25,29,30,38)(26,33,31,37)(27,34,35,40)(28,36,32,39)Polycyclic presentation:
2^9|a^2=hi,b^2=fg,c^2=fghi,d^2=gh,e^2=hi,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, 1824357);