Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #138698 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
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 | C23 | #463 |
| 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 | |
316 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 316 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)(3,7)(6,8)(9,10,14,19,16,21,13,31,15,20,12,28,30,35,29,17)(11,22,26,18,27,34,25,38,23,40,24,37,33,36,32,39);(1,2,3,5,4,6,7,8)(10,17,20,31)(11,23)(12,29)(13,14)(16,30)(18,36,37,34)(19,21,28,35)(22,38,40,39)(24,25)(26,32);(2,6)(5,8)(9,11,15,23)(10,18,20,37)(12,25,14,32)(13,24,29,26)(16,33,30,27)(17,34,31,36)(19,22,28,40)(21,39,35,38)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,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 | #72309607, #72309628, #72309635, … |
| maximal normal subgroup of | #72309607, #72309628, #72309635, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45895);