Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395166 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1376 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,4)(9,10,13,16,14,11,18,17)(12,20,21,15,22,23,24,19);(5,6)(7,8)(9,11)(10,14)(12,19)(13,16)(15,22)(17,18)(20,24)(21,23);(2,4)(6,8);(9,12,13,21,14,22,18,24)(10,15,17,20,11,19,16,23)Polycyclic presentation:
2^9|a^2=ef,d^2=h,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,e^b=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72321703, #72321728, #72321742, … |
| maximal normal subgroup of | #72321703, #72321728, #72321742, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 319525);