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 | #1271 |
| 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)(5,6,9,13,10,15,22,14)(7,16,18,32,19,29,34,33)(8,20,23,12,24,30,36,31)(11,25,26,21,28,35,27,17);(5,7)(6,11,14,27,15,28,13,26)(8,21,24,17)(9,18)(10,19)(12,29,30,33,31,16,20,32)(22,34)(23,35,36,25)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fi,d^2=i,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, 589);