Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7234333 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1515 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,6)(3,7,4,8)(9,10,13,16)(11,17,18,14)(12,20,21,15)(19,22,24,23)(25,26,29,33,28,32,37,31)(27,34,36,39,35,30,38,40);(2,6)(3,4)(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,23)(21,24)(25,27,28,35)(26,30,32,34)(29,36,37,38)(31,39,33,40);(1,2)(3,8)(4,7)(5,6)(26,31)(29,37)(30,39)(32,33)(34,40)(36,38);(1,3,5,4)(2,7,6,8)(11,18)(14,17)(19,24)(22,23);(1,4,5,3)(2,7,6,8)(9,12,13,21)(10,15,16,20)(11,19,18,24)(14,22,17,23)(26,32)(30,34)(31,33)(39,40)Polycyclic presentation:
2^9|a^2=fghi,b^2=i,d^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7198409);