Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509097 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#443 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
785 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,15,12,17,11,30)(7,18,20,31,22,32,27,13)(8,23,25,36,26,16,10,28)(14,29,33,19,35,24,34,21);(1,2)(3,4)(5,7)(6,13)(8,24)(9,27)(10,21)(11,20)(12,22)(14,16)(15,32)(17,31)(18,30)(19,25)(23,35)(26,29)(28,34)(33,36);(2,4)(5,7)(6,14)(8,19)(9,20)(10,29)(11,27)(12,22)(13,16)(15,33)(17,35)(18,28)(21,26)(23,31)(24,25)(30,34)(32,36);(5,8)(6,15)(7,19)(9,10)(11,25)(12,26)(13,31)(14,33)(16,23)(17,30)(20,29)(21,22)(24,27)(34,35)Polycyclic presentation:
2^9|a^2=e,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^c=fh,h^c=hi,e^d=ei,f^d=fh,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 416402);