Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6636816 |
| 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 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1268 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)(9,10,13,16)(11,17,19,14)(12,20,22,15)(18,23,21,24)(25,26)(27,33)(28,30)(29,37)(31,35)(32,36)(34,39)(38,40);(1,3)(2,5)(4,7)(6,8)(9,11,13,19)(10,14,16,17)(12,21,22,18)(15,24,20,23)(26,30)(27,34)(32,37)(35,40);(3,7)(5,8)(25,27,29,35,28,34,36,40)(26,31,32,33,30,38,37,39);(3,7)(5,8)(9,12,13,22)(10,15,16,20)(11,18,19,21)(14,23,17,24)(26,30)(27,34)(32,37)(35,40);(2,6)(5,8)(10,16)(14,17)(15,20)(23,24)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=g,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6545427);