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×#117 | C23 |
| prime radical | #392 | C2×C4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
617 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 617 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) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,7,5,3,8,6)(9,10)(11,20,23,18)(12,24)(13,17,29,35)(14,19)(15,22)(16,26,33,21)(25,38,39,30)(27,36)(28,31)(32,37)(34,40);(1,3)(2,5)(4,6)(7,8)(9,11,14,23)(10,15,19,31)(12,21,27,26)(13,22,29,28)(16,30,33,38)(17,20,35,18)(24,32,36,34)(25,37,39,40);(4,8)(6,7)(9,12,14,27)(10,16)(11,21,23,26)(13,25,29,39)(15,30)(17,34)(18,36)(19,33)(20,24)(22,37,28,40)(31,38)(32,35)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=gh,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,d^b=dh,e^b=eg,f^b=fgh,d^c=dgi,e^c=ei,f^c=fgi,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, 38816);