Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500596 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#493 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #1726 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
557 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 557 conjugacy classes of subgroups — too many to draw. Only its 144 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 — 4 generators on 40 points:
(10,17)(11,23)(12,29)(13,15)(14,30)(18,35)(19,34)(21,22)(24,39)(25,28)(26,38)(27,32)(33,36)(37,40);(1,2,4,6)(3,7,8,5)(9,10,13,17,16,22,15,21)(11,24,27,37,28,33,26,18)(12,19,29,20,30,34,14,31)(23,35,38,36,25,40,32,39);(1,3,4,8)(2,5,6,7)(9,11)(10,18)(12,25)(13,26)(14,32)(15,27)(16,28)(17,33)(19,36)(20,35)(21,24)(22,37)(23,30)(29,38)(31,40)(34,39);(9,12,15,14,16,30,13,29)(10,19,21,31,22,34,17,20)(11,25,27,32,28,23,26,38)(18,36,24,40,37,39,33,35)Polycyclic presentation:
2^9|b^2=g,c^2=gh,d^2=ei,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,f^c=fh,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72239798, #72239816, #72239821, … |
| maximal normal subgroup of | #72239798, #72239816, #72239821, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 414773);