Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476293 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
996 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,11,16,12,9,10,15)(7,17,21,13,22,30,20,19)(8,23,25,28,27,32,26,14)(18,33,34,36,35,29,24,31);(5,7,12,22)(6,13,9,19)(8,24,27,34)(10,21,11,20)(14,31,28,36)(15,30,16,17)(18,25,35,26)(23,29,32,33);(5,8,10,25,12,27,11,26)(6,14,15,23,9,28,16,32)(7,18,20,34,22,35,21,24)(13,29,17,31,19,33,30,36);(2,4)(5,9)(6,12)(7,19)(8,23)(10,15)(11,16)(13,22)(14,25)(17,20)(18,31)(21,30)(24,33)(26,28)(27,32)(29,34)(35,36)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=def,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383683);