Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476613 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2826 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
650 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,15,12,16,11,9)(7,17,19,13,21,30,20,31)(8,22,28,24,26,27,25,14)(18,29,35,36,23,33,34,32);(5,7,11,20,12,21,10,19)(6,13,15,30,16,31,9,17)(8,23,25,35,26,18,28,34)(14,29,22,36,24,33,27,32);(5,8,11,25,12,26,10,28)(6,14,15,22,16,24,9,27)(7,18,20,34,21,23,19,35)(13,29,30,36,31,33,17,32);(2,4)(5,9,10,16,12,15,11,6)(7,17,19,31,21,30,20,13)(8,24,25,27,26,14,28,22)(18,33,34,32,23,29,35,36)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=f,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=def,d^c=de,e^c=ei,h^c=hi,g^d=gi,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386259);