Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #132501 |
| 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 | |
557 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 557 conjugacy classes of subgroups — too many to draw. Only its 76 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)(11,20,24,30)(12,25,27,33)(13,18,28,17)(14,19)(15,22)(16,26)(21,35)(23,31)(29,38)(32,40)(34,36,39,37);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15,19,31)(12,21)(13,22)(14,24)(16,29,32,35)(17,20,18,30)(23,28)(25,39,33,34)(26,36)(27,38)(37,40);(3,8)(5,7)(9,12,14,27)(10,16,19,32)(11,21,24,38)(13,26,28,40)(15,29,31,35)(17,33,18,25)(20,34,30,39)(22,36,23,37)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=i,d^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,d^b=dh,e^b=eghi,f^b=fghi,d^c=dgh,f^c=fgi,e^d=ei,f^d=fi,f^e=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 39882);