Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139440 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1898 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×#117 | C23 |
| prime radical | #25734 | C2 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#535 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
668 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 668 conjugacy classes of subgroups — too many to draw. Only its 68 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:
(1,2,6,8,5,3,4,7)(9,10,14,19)(11,20,23,36)(12,24)(13,16,28,18)(15,25)(17,27)(21,31)(22,34,38,35)(26,33)(29,39)(30,37)(32,40);(1,3,5,2)(4,7,6,8)(9,11,12,25)(10,15,24,36)(13,22,26,39)(14,23,27,21)(16,29,33,35)(17,20,19,31)(18,30,32,34)(28,38,40,37);(4,6)(7,8)(10,16)(13,28)(15,29)(17,32)(18,19)(20,34)(22,38)(24,33)(26,40)(30,31)(35,36)(37,39)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,d^2=ghi,e^2=i,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eghi,g^a=gi,h^a=hi,d^b=dh,e^b=egh,f^b=fhi,g^b=gi,h^b=hi,d^c=dgh,e^c=ei,f^c=fgh,f^d=fi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72283973, #72284078, #72284217, … |
| maximal normal subgroup of | #72283973, #72284078, #72284217, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46708);