Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500843 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#466 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #525 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #2929 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
511 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 511 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,6)(3,5,7,8)(10,17)(11,23)(12,27)(13,20)(14,15)(18,28)(19,34)(21,32)(22,31)(24,38)(25,40)(26,29)(30,36)(37,39);(1,3)(2,5)(4,7)(6,8)(9,10,18,31,15,22,13,17,16,12,20,32,14,21,28,27)(11,19,33,38,23,36,25,30,26,24,35,34,29,37,40,39);(3,7)(5,8)(9,11)(10,19)(12,24)(13,25)(14,29)(15,23)(16,26)(17,30)(18,33)(20,35)(21,37)(22,36)(27,39)(28,40)(31,38)(32,34);(9,12,18,32,15,21,13,27,16,10,20,31,14,22,28,17)(11,24,33,34,23,37,25,39,26,19,35,38,29,36,40,30)Polycyclic presentation:
2^9|a^2=g,b^2=ehi,d^2=ehi,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,h^a=hi,c^b=cg,f^c=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72321154, #72321450, #72321458, … |
| maximal normal subgroup of | #72321154, #72321450, #72321458, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 412898);