Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135037 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1804 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C22×D8 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1450 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
296 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 296 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5)(3,7)(6,8)(9,10)(11,21,24,38)(12,18)(13,17)(14,30)(15,20)(16,22,33,25)(19,28)(23,36,27,32)(26,34)(29,35)(31,40,37,39);(1,2,3,5,4,6,7,8)(9,11)(10,16,20,33)(12,22)(13,27)(14,25)(15,24)(17,31,35,37)(18,21,30,38)(19,36,34,32)(23,29)(26,40)(28,39);(2,6)(5,8)(10,17)(12,14)(13,28)(16,31)(18,19)(20,35)(21,36)(22,25)(23,40)(26,29)(27,39)(30,34)(32,38)(33,37)Polycyclic presentation:
2^9|a^2=f,b^2=dh,d^2=gi,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=cg,e^b=ei,f^b=fi,d^c=di,e^c=eh,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42233);