Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C4 | C24 |
| commutator | C23 | C22×C42 |
| socle, prime radical | C24 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
926 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)(8,12)(9,10)(13,14)(15,19)(16,18)(17,20)(21,22,26,30,25,29,27,23)(24,32,34,31,33,28,35,36);(6,10)(7,8)(15,20)(17,19)(21,23,27,29,25,30,26,22)(24,31,35,32,33,36,34,28);(5,7,9,8)(6,11,10,12)(13,15,16,20)(14,17,18,19)(21,24,25,33)(22,28,29,32)(23,31,30,36)(26,34,27,35);(5,8,9,7)(6,11,10,12)(14,18)(17,19)(24,33)(28,32)(31,36)(34,35)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=ef,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cefi,d^b=de,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264511);