Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #845297 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2407 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3504 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(10,14)(11,16)(17,18)(19,20)(21,22)(23,24);(6,8)(10,14)(11,16)(18,22)(19,23);(5,6)(7,8)(9,10,13,14)(11,15,16,12)(17,19,21,23)(18,20,22,24);(1,2)(3,4)(9,11)(10,15)(12,14)(13,16);(9,12,13,15)(10,11,14,16)(17,20)(18,23)(19,22)(21,24)Polycyclic presentation:
2^9|c^2=fg,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 752813);