Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #137497 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1879 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #3147 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1770 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
450 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 450 conjugacy classes of subgroups — too many to draw. Only its 92 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)(3,5)(4,7)(6,8)(9,10,14,19,15,20,13,31,16,21,12,28,30,35,29,17)(11,22,25,18,26,34,24,36,27,39,23,37,33,40,32,38);(10,17,21,31)(12,29)(13,14)(15,30)(18,34,37,40)(19,20,28,35)(22,38,39,36)(23,32)(24,25)(26,33);(2,5)(3,8)(6,7)(9,11,16,27)(10,18,21,37)(12,24,14,32)(13,23,29,25)(15,33,30,26)(17,34,31,40)(19,22,28,39)(20,38,35,36)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=i,d^2=g,e^2=gh,f^2=g,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,d^b=dg,e^b=egi,f^b=fg,g^b=gi,d^c=dgi,e^c=eghi,f^c=fgi,g^c=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72309410, #72309414, #72309416, … |
| maximal normal subgroup of | #72309410, #72309414, #72309416, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44743);