Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #34936 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3274 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(6,8)(9,10,15,17,14,19,12,22)(11,16,20,23,21,13,18,24);(1,2,3,4)(9,11)(10,16)(12,18)(13,19)(14,21)(15,20)(17,23)(22,24);(5,6,7,8)(9,12,14,15)(10,17,19,22)(11,20,21,18)(13,23,16,24);(9,13,15,24,14,16,12,23)(10,18,17,11,19,20,22,21);(11,21)(13,16)(18,20)(23,24)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=g,d^2=h,h^2=i,b^a=bf,c^a=cg,c^b=ci,e^b=ei,e^d=ei128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6280914);