Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17392 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3186 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,9)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16)(17,18,20,24,21,23,29,31)(19,25,26,32,27,22,28,30);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15);(3,10)(6,9)(11,16)(13,15)(17,19,21,27)(18,22,23,25)(20,28,29,26)(24,32,31,30);(2,7)(6,9)(8,14)(13,15);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)Polycyclic presentation:
2^9|a^2=h,c^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cf,h^c=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6281786);