Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C22×D4 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1468 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,20)(19,24)(21,23)(25,26,32,35,31,36,30,29)(27,37,28,39,38,33,40,34);(1,3,4,7)(2,5,6,8)(9,11)(10,15)(12,21)(13,19)(14,20)(16,24)(17,18)(22,23)(25,27,31,38)(26,33,36,37)(28,30,40,32)(29,39,35,34);(2,6)(3,7)(25,28)(26,34)(27,30)(29,33)(31,40)(32,38)(35,37)(36,39);(2,6)(5,8)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)(25,29,30,36,31,35,32,26)(27,34,40,33,38,39,28,37);(9,13,14,12)(10,16,17,22)(11,19,20,21)(15,24,18,23)(25,30,31,32)(26,35,36,29)(27,28,38,40)(33,39,37,34)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=fh,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cgi,d^b=df,e^b=ei,d^c=di,e^c=ei,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6330535);