Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7221 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #340 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C2×#340 |
| prime radical | C24 | #107 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
407 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,10,6,9,4,8)(3,11,12,16,13,7,14,15)(17,18,21,25)(19,20,27,29)(22,24,30,32)(23,31,28,26);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,19,22,26,23,24,29,25,21,27,30,31,28,32,20,18);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,15,11,16)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=h,d^2=ghi,e^2=h,f^2=gh,g^2=i,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,d^b=di,e^b=eh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34643);