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 | |
730 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)(7,17)(8,22,25,29)(9,14)(10,15)(11,16)(12,19)(13,24,30,23)(18,32,35,26)(20,28)(21,27)(31,34,36,33);(1,2)(3,4)(5,7,9,19,11,21,10,20)(6,12,15,17,16,28,14,27)(8,18,23,33,25,35,24,34)(13,26,22,31,30,32,29,36);(2,4)(5,8,9,23,11,25,10,24)(6,13,15,22,16,30,14,29)(7,18,19,33,21,35,20,34)(12,26,17,31,28,32,27,36);(6,14,16,15)(12,27,28,17)(13,29,30,22)(26,36,32,31)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=e,d^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,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, 268443);