Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437537 |
| 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 | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2714 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)(11,22)(12,18)(13,19)(14,32)(15,21)(16,20)(17,26)(23,34)(24,33)(25,35)(27,40)(28,39)(29,38)(30,36)(31,37);(6,7)(9,11,15,26)(10,17,21,22)(12,28,13,30)(14,27,16,25)(18,36,19,39)(20,40,32,35)(23,31,24,29)(33,37,34,38);(1,2,3,4)(5,6,8,7)(9,12,15,13)(10,18,21,19)(11,23,26,24)(14,29,16,31)(17,33,22,34)(20,37,32,38)(25,28,27,30)(35,39,40,36);(5,7,8,6)(9,13,15,12)(10,19,21,18)(11,24,26,23)(14,31,16,29)(17,34,22,33)(20,38,32,37)(25,30,27,28)(35,36,40,39);(9,14,15,16)(10,20,21,32)(11,25,26,27)(12,29,13,31)(17,35,22,40)(18,37,19,38)(23,28,24,30)(33,39,34,36)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=fg,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,e^b=ei,h^b=hi,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, 6351378);