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 | |
143 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 143 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)(10,13,14,15)(17,18,20,27,23,30,33,42,24,31,36,26,22,29,39,45,21,28,38,46,34,44,48,25,40,41,19,32,35,47,37,43);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,14)(13,16)(18,25)(19,33)(20,37)(22,34)(23,35)(24,40)(26,29)(27,44)(28,42)(30,46)(31,45)(32,47)(36,48)(38,39)(41,43)Polycyclic presentation:
2^9|a^2=d,b^2=ei,c^2=fh,d^2=g,f^2=hi,g^2=h,h^2=i,b^a=bc,c^a=ce,c^b=cf,d^b=def,f^b=fh,g^b=ghi,h^b=hi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1128);