Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437608 |
| 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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2116 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10,15,20)(11,16,17,24)(12,21,23,22)(13,18,14,19);(2,5)(6,8)(9,11)(10,16)(12,19)(13,21)(14,22)(15,17)(18,23)(20,24);(9,12,15,23)(10,17,20,11)(13,16,14,24)(18,22,19,21);(9,10,15,20)(11,12,17,23)(13,19,14,18)(16,21,24,22);(1,3,4,7)(2,6,5,8)(9,13,15,14)(10,18,20,19)(11,21,17,22)(12,16,23,24)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=i,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,h^b=hi,g^c=gi,h^c=hi,e^d=ei,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346963);