Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476707 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2826 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1005 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,15,11,16,10,13)(7,17,20,12,18,28,26,29)(8,21,23,31,25,30,24,14)(19,33,35,32,22,27,34,36);(1,2)(3,4)(5,7,9,20,11,18,10,26)(6,12,13,17,16,29,15,28)(8,22,23,34,25,19,24,35)(14,27,30,32,31,33,21,36);(2,4)(6,13)(7,18)(8,23)(9,10)(12,17)(14,31)(15,16)(19,34)(22,35)(24,25)(27,33)(28,29);(5,8,9,23,11,25,10,24)(6,14,13,30,16,31,15,21)(7,19,20,35,18,22,26,34)(12,27,17,32,29,33,28,36)Polycyclic presentation:
2^9|a^2=f,b^2=e,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=def,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383903);