Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23912 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#385 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1616 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,14,11,15,9,23)(7,12,17,28,18,27,24,26)(8,19,22,30,20,34,21,13)(16,31,33,25,32,36,35,29);(2,4)(5,7,10,17,11,18,9,24)(6,12,14,28,15,27,23,26)(8,16,22,33,20,32,21,35)(13,29,19,31,30,25,34,36);(5,8)(6,13,15,30)(7,16)(9,21)(10,22)(11,20)(12,25,27,29)(14,19,23,34)(17,33)(18,32)(24,35)(26,31,28,36);(2,4)(8,20)(13,30)(16,32)(19,34)(21,22)(25,29)(31,36)(33,35)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275253);