Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574086 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1958 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,8)(4,6)(5,7)(9,10,14,21)(11,23,25,17)(12,19,29,39)(13,20,31,27)(15,33,16,22)(18,35,38,30)(24,36,40,32)(26,37,34,28);(1,3,5,4)(2,6,7,8)(9,11,14,25)(10,17,21,23)(12,27,29,20)(13,19,31,39)(15,26,16,34)(18,32,38,36)(22,37,33,28)(24,35,40,30);(9,12,16,35,14,29,15,30)(10,18,33,39,21,38,22,19)(11,24,34,27,25,40,26,20)(13,28,36,23,31,37,32,17);(1,4,5,3)(2,6,7,8)(9,12,14,29)(10,19,21,39)(11,20,25,27)(13,23,31,17)(15,30,16,35)(18,22,38,33)(24,26,40,34)(28,36,37,32);(9,13)(10,20)(11,18)(12,28)(14,31)(15,32)(16,36)(17,30)(19,26)(21,27)(22,40)(23,35)(24,33)(25,38)(29,37)(34,39)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6491549);