Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1339 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #327 |
| commutator | C2×C8 | C4×C8 |
| Frattini, cosocle | C22×#97 | C22 |
| socle | C22 | #1270 |
| prime radical | C24 | #104 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
360 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 36 points:
(1,2,3,4)(5,6)(7,15)(8,19)(9,24)(10,14)(11,16)(12,23)(13,21)(17,31)(18,27)(20,26)(22,28)(25,35)(29,36)(30,34)(32,33);(5,7)(6,11,13,25,14,26,24,30)(8,20,22,16)(9,17)(10,18)(12,27,28,32,29,15,19,31)(21,33)(23,35,36,34)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fi,e^2=h,f^2=i,b^a=bc,c^a=cf,e^a=efgi,f^a=fi,h^a=hi,c^b=cg,g^b=gi,e^c=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 588);