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 | #2619 | C22 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
637 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 637 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)(11,19)(12,23,25,15)(13,26,27,17)(14,18)(16,21,33,28)(20,37,39,36)(22,34)(24,31)(29,35)(30,38)(32,40);(1,3)(2,5)(4,6)(7,8)(10,15)(13,27)(16,30)(17,31)(18,23)(19,36)(21,28)(24,29)(26,35)(32,33)(34,37)(38,40);(4,8)(6,7)(9,11,14,22)(10,16)(12,20,25,39)(13,28,27,21)(15,30)(17,34)(18,33)(19,26)(23,32)(24,40,29,38)(31,37)(35,36)Polycyclic presentation:
2^9|a^2=f,c^2=ghi,d^2=h,f^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, 38771);