Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667604 |
| 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 | C22×D8 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1391 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)(3,6)(4,8)(5,7)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26,30,31)(27,34,37,32)(28,33,29,39)(35,38,40,36);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27,30,37)(26,32,31,34)(28,38,29,36)(33,35,39,40);(1,4,5,3)(2,6,7,8)(25,28)(26,33)(27,35)(29,30)(31,39)(32,36)(34,38)(37,40);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,29,30,28)(26,33,31,39)(27,36,37,38)(32,35,34,40);(11,19)(14,17)(18,24)(22,23)(26,31)(27,32)(33,39)(34,37)(35,36)(38,40)Polycyclic presentation:
2^9|a^2=f,b^2=gi,c^2=g,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72308700, #72308731, #72308959, … |
| maximal normal subgroup of | #72308700, #72308731, #72308959, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6598114);