Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #459 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1879 |
| prime radical | C2×#135 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
408 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 408 conjugacy classes of subgroups — too many to draw. Only its 82 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:
(1,2,4,7,5,8,3,6)(9,10,14,20,15,21,13,19,16,22,30,38,23,33,31,39)(11,18,24,17,25,34,29,35,26,37,32,36,28,40,12,27);(2,6,8,7)(9,11,23,25,16,26,15,28)(10,17,33,35,22,36,21,27)(12,13,24,14,29,31,32,30)(18,19,34,20,37,39,40,38);(9,12)(10,18)(11,13)(14,25)(15,32)(16,29)(17,19)(20,35)(21,40)(22,37)(23,24)(26,31)(27,38)(28,30)(33,34)(36,39)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,d^2=ghi,e^2=hi,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=eh,f^b=fh,h^b=hi,d^c=dh,e^c=eh,f^c=fh,h^c=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45916);