Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7221 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #340 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | C2×#340 |
| prime radical | C24 | C2×Q16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
407 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18,21,27,24,29,23,40)(19,30,32,42,35,44,34,48)(20,36,38,45,39,28,22,26)(25,33,41,37,43,47,46,31);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19,22,33,23,34,39,31,24,35,38,47,21,32,20,37)(18,25,28,42,27,41,26,44,29,43,36,48,40,46,45,30);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=h,d^2=ghi,e^2=i,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, 34562);