Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| 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 | #392 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1283 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)(3,7)(4,5)(6,8)(9,10)(11,17)(12,16)(13,15)(14,18)(19,23)(20,22)(21,24)(25,26,29,34,30,35,38,40)(27,31,37,39,36,33,28,32);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,19)(13,18)(15,17)(16,21)(20,24)(22,23)(25,27)(26,31)(28,38)(29,37)(30,36)(32,40)(33,35)(34,39);(2,5)(6,7)(11,18)(14,17)(19,24)(21,23)(25,28)(26,32)(27,29)(30,37)(31,34)(33,40)(35,39)(36,38);(1,3,4,8)(2,6,5,7)(10,15)(11,18)(16,22)(19,24)(25,27)(26,33)(28,38)(29,37)(30,36)(31,35)(32,34)(39,40);(9,12,13,20)(10,16,15,22)(11,19,18,24)(14,21,17,23)(27,36)(28,37)(31,33)(32,39)Polycyclic presentation:
2^9|a^2=h,d^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=cfgi,d^b=df,e^b=ei,d^c=di,e^c=ei,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6315027);