Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139440 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×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 | C2×#535 | C22 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#535 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
586 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 586 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) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,5,9,6,8,4,14)(3,10,13,15,12,16,11,7)(17,18,22,27)(19,28)(20,25)(21,26,37,42)(23,29,40,30)(24,36)(31,38)(32,45)(33,41)(34,43)(35,48)(39,46,44,47);(1,3)(2,7,8,15)(4,11)(5,13)(6,12)(9,10,14,16)(17,19,20,29)(18,23,25,28)(21,35,33,47)(22,32,34,30)(24,38,42,39)(26,44,41,48)(27,40,43,45)(31,36,46,37);(2,8)(4,5)(7,15)(11,13)(17,20)(18,24)(19,29)(21,36)(22,34)(23,38)(25,42)(26,43)(27,41)(28,39)(30,32)(31,47)(33,37)(35,46)(40,48)(44,45)Polycyclic presentation:
2^9|a^2=f,b^2=df,d^2=gh,e^2=i,f^2=ghi,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=fh,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 | #72284041, #72284078, #72284183, … |
| maximal normal subgroup of | #72284041, #72284078, #72284183, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46686);