Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #118246 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1439 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
897 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)(3,5)(4,6)(7,8)(9,10)(11,19)(12,23,25,16)(13,26)(14,18)(15,21)(17,27)(20,32,35,40)(22,33)(24,31,38,37)(28,39,30,34)(29,36);(1,2)(3,7,8,5)(4,6)(9,11,13,21,14,22,27,36)(10,15,17,19,18,29,26,33)(12,20,24,34,25,35,38,39)(16,30,31,40,23,28,37,32);(1,3)(2,5)(4,8)(6,7)(9,12)(10,16)(11,20)(13,24)(14,25)(15,28)(17,31)(18,23)(19,32)(21,34)(22,35)(26,37)(27,38)(29,30)(33,40)(36,39)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,e^b=eh,f^b=fi,d^c=di,g^c=gh56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 25442);