Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144129 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2011 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #3147 | C22 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
454 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 454 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,5,3)(4,7,6,8)(9,10)(11,20)(12,25)(13,31)(14,19)(15,22)(16,29)(17,35)(18,28)(21,24)(23,37)(26,30)(27,38)(32,36)(33,40)(34,39);(2,7)(3,8)(4,6)(9,10)(11,21)(12,26)(13,31)(14,19)(15,22)(16,35)(17,29)(18,28)(20,24)(23,36)(25,30)(27,38)(32,37)(33,40)(34,39);(1,3)(2,5)(4,8)(6,7)(9,11,14,24)(10,15,19,34)(12,27,30,40)(13,32,28,23)(16,36,17,37)(18,38,31,33)(20,26,21,25)(22,35,39,29)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=g,e^2=hi,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fghi,h^a=hi,c^b=cf,d^b=dg,e^b=ehi,f^b=fgh,h^b=hi,d^c=dg,e^c=eh,f^c=fgh,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51342);