Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144119 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2011 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #3147 | C22 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
507 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 507 conjugacy classes of subgroups — too many to draw. Only its 100 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) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,6)(4,7)(5,8)(9,10,15,12)(11,19,22,39)(13,27,32,29)(14,26,30,18)(16,34,24,21)(17,36,25,31)(20,33,37,23)(28,38,40,35);(2,6)(3,4)(7,8)(9,11)(10,16)(12,24)(13,28)(14,33)(15,22)(17,37)(18,38)(19,27)(20,25)(21,31)(23,30)(26,35)(29,39)(32,40)(34,36);(2,7)(3,4)(6,8)(9,12,15,10)(11,20,22,37)(13,29,32,27)(14,18,30,26)(16,28,24,40)(17,31,25,36)(19,23,39,33)(21,38,34,35)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=gi,e^2=h,f^2=ghi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgh,g^a=gi,c^b=cf,d^b=dg,e^b=ehi,f^b=fgh,g^b=gi,d^c=dg,e^c=eh,f^c=fgh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51317);