Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #141286 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1982 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #460 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
334 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 334 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,3)(4,5)(7,8)(9,10,14,19,29,32,27,18,13,17,28,36,15,20,12,26)(11,21,24,16,31,34,23,38,22,39,40,33,25,37,30,35);(1,2,4,3,6,8,5,7)(9,11,15,31,13,22,29,25)(10,16,32,35,17,33,20,38)(12,24,28,23,27,40,14,30)(18,37,36,39,26,34,19,21);(1,3,4,8,6,7,5,2)(10,17)(11,22)(18,26)(19,36)(20,32)(23,30)(24,40)(25,31)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,c^2=eg,d^2=ghi,e^2=gi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,c^b=cg,d^b=dh,f^b=fh,h^b=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72309487, #72309494, #72309508, … |
| maximal normal subgroup of | #72309487, #72309494, #72309508, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48504);