Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C4 | C24 |
| commutator, socle, prime radical | C24 | C23×C4 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
994 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,11)(8,9)(10,12)(21,22,27,23,26,30,25,29)(24,28,32,31,33,35,34,36);(6,9)(10,11)(13,14,16,18)(15,19,20,17)(21,23,27,30,26,29,25,22)(24,31,32,35,33,36,34,28);(2,4)(13,15,16,20)(14,17,18,19)(21,24)(22,28)(23,31)(25,34)(26,33)(27,32)(29,36)(30,35);(5,7,8,12)(6,10,9,11)(21,25,26,27)(22,29,30,23)(24,32,33,34)(28,31,35,36)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=g,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 277208);