Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133329 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C22×D4 | C22×C4 |
| Frattini, cosocle | C2×C4×D4 | C23 |
| prime radical | C2×H(3,𝔽4) | C4 |
| LCS | G ▷ C22×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
688 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 688 conjugacy classes of subgroups — too many to draw. Only its 80 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:
(1,2,3,6,4,5,7,8)(9,10)(11,19,22,15)(12,23)(13,27)(14,17)(16,26)(18,29)(20,36,24,40)(21,31,39,34)(25,35)(28,33)(30,37,32,38);(2,5)(3,7)(9,11)(10,15)(12,24)(13,21)(14,22)(16,32)(17,19)(18,34)(20,25)(23,40)(26,38)(27,31)(28,37)(29,39)(30,33)(35,36);(9,12,14,25)(10,16,17,33)(11,20,22,24)(13,28,29,26)(15,30,19,32)(18,35,27,23)(21,38,39,37)(31,36,34,40)Polycyclic presentation:
2^9|a^2=f,c^2=i,e^2=i,f^2=gi,b^a=bd,c^a=ce,d^a=dg,e^a=ei,h^a=hi,c^b=ci,e^b=ehi,f^b=fg,d^c=dh,e^c=ei,e^d=ei44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 40747);