Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2713 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,4)(5,6,8,7)(9,10,13,17)(11,19,22,15)(12,16,25,33)(14,18,27,35)(20,37,38,29)(21,31,23,30)(24,40,39,32)(26,34,28,36);(6,7)(9,11,13,22)(10,15,17,19)(12,24,25,39)(14,21,27,23)(16,32,33,40)(18,30,35,31)(20,28,38,26)(29,36,37,34);(1,2,3,4)(5,6,8,7)(9,12,13,25)(10,16,17,33)(11,20,22,38)(14,28,27,26)(15,29,19,37)(18,36,35,34)(21,24,23,39)(30,32,31,40);(5,7,8,6)(11,21)(14,27)(15,30)(18,35)(19,31)(20,24)(22,23)(26,28)(29,32)(34,36)(37,40)(38,39);(12,25)(16,33)(20,38)(24,39)(26,28)(29,37)(32,40)(34,36)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fg,d^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351387);