Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #189517 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#117 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2283 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #1729 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
750 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,15,11,16,9,12)(7,17,19,29,20,13,25,26)(8,21,23,30,24,27,22,14)(18,32,33,31,34,35,36,28);(1,2)(3,4)(6,12,16,15)(13,26,17,29)(14,27,30,21)(28,32,31,35);(2,4)(5,7,9,19,11,20,10,25)(6,13,12,26,16,17,15,29)(8,18,22,33,24,34,23,36)(14,28,27,32,30,31,21,35);(5,8,9,22,11,24,10,23)(6,14,15,21,16,30,12,27)(7,18,19,33,20,34,25,36)(13,28,29,35,17,31,26,32)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=ef,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 268166);