Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29730 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C2×#121 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
785 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,13)(8,12)(9,14)(10,15)(11,16)(17,20)(18,19)(21,22,24,28,25,27,31,34)(23,26,30,33,29,32,35,36);(5,7,9,8,11,18,10,17)(6,12,15,13,16,20,14,19)(21,23,24,30,25,29,31,35)(22,26,28,33,27,32,34,36);(2,4)(5,8,9,18,11,17,10,7)(6,13,15,20,16,19,14,12)(23,29)(26,32)(30,35)(33,36);(5,9,11,10)(6,14,16,15)(7,8,18,17)(12,19,20,13)(22,27)(26,32)(28,34)(33,36)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=e,d^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,c^b=chi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 109782);