Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D4×D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#508 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1042 conjugacy classes of subgroups; only the 238 normal subgroups are shown.
This group has 1042 conjugacy classes of subgroups — too many to draw. Only its 238 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) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,4)(5,6,7,8)(10,16)(12,24)(13,26)(14,28)(17,30)(18,25)(19,31)(20,34)(21,36)(22,38)(27,29)(32,35)(33,40)(37,39);(9,10)(11,17)(12,25)(13,27)(14,29)(15,18)(16,26)(19,28)(20,35)(21,37)(22,39)(23,32)(24,31)(30,36)(33,38)(34,40);(2,4)(9,11,15,23)(10,17,18,32)(12,20,26,36)(13,21,24,34)(14,22,28,38)(16,30,25,35)(19,33,29,39)(27,37,31,40);(9,12)(10,18)(11,20)(13,28)(14,24)(15,26)(16,31)(17,32)(21,38)(22,34)(23,36)(25,27)(30,40)(35,37)Polycyclic presentation:
2^9|a^2=g,c^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eh,h^a=hi,d^b=de,e^b=eh,h^b=hi,e^d=eh,h^d=hi110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404640);