Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454426 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1019 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,5)(4,6)(7,8)(9,10,12,13)(11,15,16,14)(17,18,22,25,21,20,23,26)(19,27,30,32,29,24,31,28);(1,3,4,7)(2,5,6,8)(10,13)(14,15)(17,19,23,31,21,29,22,30)(18,24,26,32,20,27,25,28);(2,6)(5,8)(9,11)(10,14)(12,16)(13,15)(17,18)(19,28)(20,21)(22,26)(23,25)(24,30)(27,31)(29,32);(11,16)(14,15)(17,20,21,18)(19,27,29,24)(22,26,23,25)(28,30,32,31)Polycyclic presentation:
2^9|a^2=eg,b^2=ef,d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,f^c=fi,g^c=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367799);