Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393881 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1656 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(6,7)(9,10)(11,16)(12,15)(13,14)(17,18,19,22,20,23,24,21);(2,4)(5,6,8,7)(10,14)(15,16)(18,21)(19,24)(22,23);(1,2,3,4)(5,6,8,7)(9,11,13,12)(10,15,14,16);(5,7,8,6)(9,12,13,11)(10,15,14,16)Polycyclic presentation:
2^9|a^2=ef,b^2=h,c^2=fg,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 317828);