Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #151127 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2177 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C4×C16 | C23 |
| prime radical | C2×D16 | C2×C4 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C82 ◁ #1311 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
135 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 135 conjugacy classes of subgroups — too many to draw. Only its 52 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:
(1,2,5,3,4,6,8,7)(9,10,13,19,15,17,11,21)(12,24,26,34,28,16,22,32)(14,30,29,20,31,38,23,36)(18,37,39,25,35,40,33,27);(2,6)(3,7)(10,16)(11,13)(12,25)(14,29)(17,24)(18,36)(19,34)(20,35)(21,32)(22,40)(23,31)(26,37)(27,28)(30,33)(38,39);(1,3,5,6,4,7,8,2)(9,11,15,13)(10,17)(12,26,28,22)(14,23,31,29)(19,21)(20,36)(25,40,27,37)(30,38)Polycyclic presentation:
2^9|a^2=e,c^2=eh,d^2=fhi,e^2=g,f^2=hi,h^2=i,b^a=bd,c^a=cgh,d^a=df,f^a=fh,h^a=hi,c^b=cg,d^b=dfi,e^b=ei,f^b=fh,h^b=hi,d^c=di56 conjugacy classes — view the full character table.
| maximal subgroup of | #72232014, #72232016, #72232019, … |
| maximal normal subgroup of | #72232014, #72232016, #72232019, … |
| maximal quotient of | #72321047, #72322783 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58323);