Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2632 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,3)(9,10,16,21)(11,17,25,38)(12,26,23,40)(13,30,22,39)(14,20,15,32)(18,28,35,29)(19,27,34,31)(24,36,33,37);(1,2,4,3)(5,6)(7,8)(9,11)(10,17)(12,27)(13,28)(14,24)(15,33)(16,25)(18,39)(19,40)(20,36)(21,38)(22,29)(23,31)(26,34)(30,35)(32,37);(9,12,16,23)(10,18,21,35)(11,22,25,13)(14,29,15,28)(17,34,38,19)(20,40,32,26)(24,31,33,27)(30,36,39,37);(6,8)(9,13,16,22)(10,19,21,34)(11,23,25,12)(14,31,15,27)(17,35,38,18)(20,39,32,30)(24,29,33,28)(26,36,40,37);(1,3,4,2)(9,14,16,15)(10,20,21,32)(11,24,25,33)(12,28,23,29)(13,31,22,27)(17,36,38,37)(18,26,35,40)(19,39,34,30)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,d^2=i,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7426382);