Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #612424 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3624 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(6,7)(10,13)(12,15)(17,18,20,22)(19,21,23,24);(1,2,4,3)(5,6,8,7)(9,10,14,13)(11,15,16,12)(19,23)(21,24);(5,6,8,7)(9,11,14,16)(10,15,13,12)(17,19,20,23)(18,21,22,24);(5,6,8,7)(9,12)(10,11)(13,16)(14,15)(17,18)(19,21)(20,22)(23,24);(1,3,4,2)(5,7,8,6)(9,13,14,10)(11,12,16,15)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=i,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 522780);