Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499266 |
| 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 | #3150 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #1726 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
520 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 520 conjugacy classes of subgroups — too many to draw. Only its 130 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:
(1,2,4,6)(3,7,8,5)(9,10)(11,21,24,31)(12,18)(13,19)(14,20)(15,17)(16,23,32,25)(22,36,39,37)(26,34)(27,35)(28,40,38,30)(29,33);(1,3,4,8)(2,5,6,7)(11,22,24,39)(12,14)(13,27)(16,30,32,40)(18,20)(19,35)(21,37,31,36)(23,38,25,28)(26,29)(33,34);(2,6)(5,7)(9,11)(10,16)(12,25)(13,28)(14,23)(15,24)(17,32)(18,31)(19,36)(20,21)(22,26)(27,39)(29,38)(30,34)(33,37)(35,40);(10,17)(11,23,24,25)(13,29)(16,31,32,21)(18,20)(22,38,39,28)(26,27)(30,36,40,37)Polycyclic presentation:
2^9|a^2=e,b^2=e,d^2=efh,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,f^b=fi,g^b=gh,h^b=hi,d^c=dh,e^c=ei,f^c=fi,g^c=ghi,h^c=hi,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72239796, #72239811, #72239836, … |
| maximal normal subgroup of | #72239796, #72239811, #72239836, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406889);