Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #131240 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C2×C22⋊C4 | C22×C4 |
| Frattini, cosocle | C2×C4×D4 | C23 |
| prime radical | C2×#120 | D4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
561 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 561 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5,6,7)(9,10)(11,19,22,31)(12,23)(13,27,29,17)(14,18)(15,21)(16,25,33,24)(20,30,38,40)(26,34)(28,32)(35,36)(37,39);(1,2)(3,7)(4,6)(5,8)(9,11,14,22)(10,15,18,32)(12,20,26,38)(13,28,29,21)(16,30,33,40)(17,31,27,19)(23,35,34,39)(24,37,25,36);(9,12)(10,16,18,33)(11,20)(13,24)(14,26)(15,30,32,40)(17,34,27,23)(19,35,31,39)(21,36)(22,38)(25,29)(28,37)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,d^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=egi,f^b=fgh,d^c=dg,e^c=ei,f^c=fg,e^d=ei,f^d=fi,f^e=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 38438);