Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7418 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #357 |
| commutator | C2×C16 | C2×C8 |
| Frattini, cosocle | C2×C4×C16 | C22 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
171 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 171 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 2 generators on 48 points:
(1,2)(3,7,8,5)(4,6)(9,10,11,13,12,14,15,16)(18,24,25,40)(19,29)(20,35)(21,31)(22,37)(26,43,27,44)(28,45,39,42)(30,36,41,38)(32,34)(33,48)(46,47);(1,3)(2,5)(4,8)(6,7)(17,18,23,25)(19,30,34,41)(20,36,21,38)(22,39,37,28)(24,29,40,32)(26,31,27,35)(33,43,46,44)(42,48,45,47)Polycyclic presentation:
2^9|a^2=d,b^2=i,c^2=fh,d^2=g,e^2=hi,f^2=hi,h^2=i,b^a=bc,c^a=ce,e^a=eh,f^a=fh,h^a=hi,c^b=cf,d^b=defh,e^b=eh,f^b=fh,h^b=hi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1112);