Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #346950 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2846 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #32911 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2212 conjugacy classes of subgroups; only the 222 normal subgroups are shown.
This group has 2212 conjugacy classes of subgroups — too many to draw. Only its 222 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:
(2,5)(3,7)(6,8)(9,10)(11,21)(12,18)(13,19)(14,17)(15,20)(16,23)(22,36)(24,31)(25,29)(26,32)(27,34)(28,33)(30,37)(35,38)(39,40);(1,2,4,6)(3,8,7,5)(9,11,15,25)(10,16,20,31)(12,22,27,38)(13,24,14,23)(17,29,19,21)(18,30,34,40)(26,39,28,37)(32,36,33,35);(10,17)(13,14)(16,29)(18,32)(19,20)(21,31)(23,24)(26,28)(30,36)(33,34)(35,40)(37,39);(9,12)(10,18)(11,22)(13,28)(14,26)(15,27)(16,30)(17,32)(19,33)(20,34)(21,35)(23,39)(24,37)(25,38)(29,36)(31,40)Polycyclic presentation:
2^9|b^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,e^b=eh,f^b=fi,g^b=gi,e^c=ei,f^c=fi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 254147);