Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6343485 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | #62811 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
4322 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7,8,5)(4,6)(9,10)(11,20,23,30)(12,17)(13,27)(14,18)(15,19)(16,22,31,28)(21,34,35,40)(24,32)(25,37)(26,33)(29,36,39,38);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,21)(13,22)(14,28)(15,23)(17,29)(18,30)(19,31)(20,27)(24,35)(25,36)(26,38)(32,39)(33,40)(34,37);(9,12)(10,17)(11,21)(13,25)(14,26)(15,24)(16,29)(18,33)(19,32)(20,34)(22,36)(23,35)(27,37)(28,38)(30,40)(31,39);(12,24)(17,32)(21,35)(25,26)(29,39)(33,37)(34,40)(36,38);(2,6)(5,7)(9,13,15,14)(10,18,19,27)(11,22,23,28)(12,25,24,26)(16,30,31,20)(17,33,32,37)(21,36,35,38)(29,40,39,34)Polycyclic presentation:
2^9|a^2=g,e^2=i,f^2=i,b^a=bf,e^a=ehi,f^a=fh,f^b=fi,g^b=ghi,d^c=di98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250828);