Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #2678 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4403 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,19)(13,16)(15,20)(17,23)(18,22)(21,24);(2,5)(6,8)(9,11,13,18)(10,14,16,22)(12,17,20,24)(15,21,19,23)(25,26)(27,33)(28,32)(29,31)(30,34)(35,40)(36,38)(37,39);(9,12,13,20)(10,15,16,19)(11,17,18,24)(14,21,22,23)(25,27)(26,30)(28,36)(29,34)(31,33)(32,40)(35,37)(38,39);(12,20)(15,19)(17,24)(21,23)(26,31)(27,34)(32,39)(35,36);(10,16)(14,22)(15,19)(21,23)(25,28,29,37)(26,32,31,39)(27,35,34,36)(30,38,33,40);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|b^2=h,c^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7614722);