Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #141233 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1982 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #3169 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
309 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 309 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,5,9,6,10,4,7)(3,11,13,16,14,15,12,8)(17,18,22,27,37,40,35,26,21,25,36,44,23,28,20,34)(19,29,32,24,39,42,30,46,31,47,48,41,33,45,38,43);(2,7,10,9)(8,15,16,11)(17,19,23,39,21,31,37,33)(18,24,40,43,25,41,28,46)(20,32,36,30,35,48,22,38)(26,45,44,47,34,42,27,29);(1,3,4,12,6,14,5,13)(2,8,9,11,10,16,7,15)(18,25)(24,41)(26,34)(27,44)(28,40)(29,47)(42,45)(43,46)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,c^2=ei,d^2=ghi,e^2=gi,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,d^b=dh,f^b=fh,h^b=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72287554, #72287608, #72287611, … |
| maximal normal subgroup of | #72287554, #72287608, #72287611, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48523);