Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #111436 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #390 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1439 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
881 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,7,8,5)(4,6)(9,10,14,19,15,20,13,18)(11,16,24,32,22,30,23,31)(12,17,27,35,26,34,28,36)(21,37,25,40,38,29,39,33);(9,11)(10,16)(12,25,26,39)(13,23)(14,24)(15,22)(17,33,34,40)(18,31)(19,32)(20,30)(21,27,38,28)(29,35,37,36);(1,3)(2,5)(4,8)(6,7)(9,12)(10,17)(11,21)(13,28)(14,27)(15,26)(16,29)(18,36)(19,35)(20,34)(22,38)(23,39)(24,25)(30,37)(31,40)(32,33)Polycyclic presentation:
2^9|a^2=g,b^2=d,f^2=h,g^2=h,b^a=bd,c^a=ce,e^a=ei,c^b=cf,e^b=eh,d^c=dh,f^c=fh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 23186);