Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437935 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3667 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(12,19)(14,20)(17,21)(22,24);(10,14,16,20)(11,17,18,21);(2,5)(6,8)(9,10)(11,15)(12,20)(13,16)(14,19)(17,22)(18,23)(21,24);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,21)(13,18)(14,22)(16,23)(17,19)(20,24);(10,16)(14,20)(15,23)(22,24)Polycyclic presentation:
2^9|b^2=f,d^2=g,h^2=i,b^a=bf,c^a=cg,h^a=hi,c^b=ch,d^b=dh,e^c=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72251976, #72255753, #72255784, … |
| maximal normal subgroup of | #72251976, #72255753, #72255784, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6356293);