Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455826 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2631 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
756 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,4)(3,6,7,8)(10,16)(11,18)(12,20)(13,14)(17,24)(19,21);(1,3,5,7)(2,6,4,8)(9,10,14,11,15,18,13,16)(12,17,22,19,20,21,23,24);(1,4,5,2)(3,6,7,8)(9,11,13,10,15,16,14,18)(12,21,22,24,20,17,23,19);(9,12,14,22,15,20,13,23)(10,17,11,19,18,21,16,24)Polycyclic presentation:
2^9|a^2=g,b^2=fg,c^2=eg,d^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,h^b=hi,d^c=de,h^c=hi,e^d=ei,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 376187);