Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23906 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
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 | C4×#120 |
| prime radical | C2×#3018 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2382 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,12,7,10,15,11,18)(8,19,21,13,22,16,26,29)(9,23,25,30,24,14,27,17)(20,32,34,28,35,31,33,36);(1,2)(3,4)(5,7,12,15,10,18,11,6)(8,16,21,29,22,19,26,13)(9,17,25,23,24,30,27,14)(20,31,34,36,35,32,33,28);(2,4)(5,8)(6,13)(7,16)(9,20)(10,22)(11,26)(12,21)(14,28)(15,29)(17,31)(18,19)(23,36)(24,35)(25,34)(27,33)(30,32);(5,9)(6,14)(7,17)(8,20)(10,24)(11,27)(12,25)(13,28)(15,23)(16,31)(18,30)(19,32)(21,34)(22,35)(26,33)(29,36)Polycyclic presentation:
2^9|a^2=h,b^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,f^b=fi,e^c=ei,f^c=fi,f^d=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271751);