Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395169 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1028 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,6)(7,8)(9,10,13,16,14,11,18,17)(12,20,22,15,21,23,24,19);(1,2)(3,4)(5,7,8,6)(12,21)(15,19)(20,23)(22,24);(2,4)(6,7)(9,11)(10,14)(12,19)(13,16)(15,21)(17,18)(20,24)(22,23);(9,12,13,22,14,21,18,24)(10,15,17,20,11,19,16,23)Polycyclic presentation:
2^9|a^2=ef,b^2=e,d^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=di,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72321674, #72321693, #72321742, … |
| maximal normal subgroup of | #72321674, #72321693, #72321742, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 319723);