Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7300948 |
| 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 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1271 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10,13,16)(11,17,19,14)(12,15,20,22)(18,23,24,21);(1,3,5,4)(2,6,8,7)(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,26)(27,33)(28,36)(29,31)(30,34)(32,37)(35,38)(39,40);(2,7)(3,4)(6,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,27)(26,30)(28,35)(29,34)(31,33)(32,38)(36,39)(37,40);(1,3,5,4)(2,6,8,7)(11,19)(14,17)(18,24)(21,23)(26,31)(27,34)(32,36)(35,40);(1,3,5,4)(2,6,8,7)(10,16)(14,17)(15,22)(21,23)(25,28,29,37)(26,32,31,36)(27,35,34,40)(30,38,33,39)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=f,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=ehi,d^c=dh,e^c=ei,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7361526);