Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #140270 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1926 |
| 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×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
260 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 84 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:
(2,7)(3,8)(4,5)(9,10)(11,19,22,35)(12,15)(13,25)(14,28)(16,21,30,37)(17,27)(18,20)(23,38,29,33)(24,39)(26,32,40,34)(31,36);(1,2,5,3,6,8,4,7)(10,15)(13,24)(16,30)(18,28)(21,37)(26,40)(27,36)(32,34);(1,3,5,8,6,7,4,2)(9,11,20,23,12,22,14,29)(10,16,28,32,15,30,18,34)(13,26,36,37,24,40,27,21)(17,33,39,35,31,38,25,19)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=dh,d^2=g,e^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=cg,e^b=ei,d^c=di,e^c=egh,f^c=fi,h^c=hi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72309482, #72309499, #72309508, … |
| maximal normal subgroup of | #72309482, #72309499, #72309508, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47481);