Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3497 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4547 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)(3,4)(9,10,15,20)(11,21,24,34)(12,17,26,32)(13,19,14,18)(16,28,30,23)(22,35,37,39)(25,33,27,31)(29,36,40,38);(5,6)(7,8)(9,11)(10,16)(12,22)(13,23)(14,28)(15,24)(17,29)(18,21)(19,34)(20,30)(25,36)(26,37)(27,38)(31,39)(32,40)(33,35);(2,4)(6,8)(9,12)(10,17)(11,22)(13,27)(14,25)(15,26)(16,29)(18,33)(19,31)(20,32)(21,35)(23,38)(24,37)(28,36)(30,40)(34,39);(5,6)(7,8)(9,13,15,14)(10,18,20,19)(11,23,24,28)(12,25,26,27)(16,21,30,34)(17,31,32,33)(22,36,37,38)(29,39,40,35);(6,8)(12,26)(17,32)(22,37)(25,27)(29,40)(31,33)(35,39)(36,38)Polycyclic presentation:
2^9|a^2=i,d^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6480825);