Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #138698 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
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 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
359 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 359 conjugacy classes of subgroups — too many to draw. Only its 90 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:
(2,5)(3,6)(7,8)(9,10,12,20,15,22,13,16,14,21,29,18,31,39,32,19)(11,23,25,17,27,35,26,37,24,38,30,36,33,34,28,40);(1,2,6,8,4,7,3,5)(10,16,21,19)(11,24)(12,13)(15,31)(17,35,36,34)(18,22,20,39)(23,40,38,37)(25,28)(26,30)(29,32);(2,7)(5,8)(9,11,14,24)(10,17,21,36)(12,28,29,26)(13,30,32,25)(15,33,31,27)(16,34,19,35)(18,38,20,23)(22,40,39,37)Polycyclic presentation:
2^9|a^2=f,b^2=dfgh,c^2=i,d^2=ghi,e^2=hi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=cg,d^b=dh,e^b=ehi,f^b=fh,h^b=hi,d^c=dhi,e^c=eh,f^c=fhi,h^c=hi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72309451, #72309461, #72309462, … |
| maximal normal subgroup of | #72309451, #72309461, #72309462, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45913);