Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #138564 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1893 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #532 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2931 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
343 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 343 conjugacy classes of subgroups — too many to draw. Only its 96 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:
(1,2,4,6,5,8,3,7)(9,10,14,20,16,21,15,19)(11,22,25,23,26,32,18,13)(12,27,30,34,31,35,29,17)(24,28,38,36,39,37,33,40);(2,6)(3,4)(7,8)(9,11,23,21,15,25,22,19,16,26,13,10,14,18,32,20)(12,24,37,34,29,38,36,35,31,39,28,17,30,33,40,27);(2,6)(3,4)(7,8)(9,12)(10,17)(11,24)(13,28)(14,30)(15,29)(16,31)(18,33)(19,35)(20,27)(21,34)(22,36)(23,37)(25,38)(26,39)(32,40)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,d^2=ghi,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dh,e^b=ehi,f^b=fh,d^c=dhi,e^c=eh,f^c=fhi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45785);