Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×C22⋊C4 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C8×C22⋊C4 |
| prime radical | #121 | C2×C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
285 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)(4,12)(5,8)(6,11)(7,13)(10,15)(14,16)(17,18,22,28,24,27,42,46,23,29,41,31,21,38,40,47)(19,30,33,34,32,25,43,20,26,35,45,37,39,44,48,36);(1,3)(2,6,8,14)(4,11)(5,10)(7,9,12,15)(13,16)(17,19)(18,25)(20,28)(21,39)(22,33)(23,26)(24,32)(27,35)(29,44)(30,38)(31,36)(34,47)(37,46)(40,48)(41,45)(42,43);(3,10)(6,14)(9,15)(11,16)(17,20,22,35,24,37,42,44,23,36,41,30,21,34,40,25)(18,26,28,45,27,39,46,48,29,19,31,33,38,32,47,43)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=g,e^2=h,h^2=i,b^a=bd,d^a=dg,f^a=fgi,c^b=cg,d^b=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35846);