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 | |
1014 conjugacy classes of subgroups; only the 238 normal subgroups are shown.
This group has 1014 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)(10,15)(11,20)(12,21)(13,22)(16,27)(17,23)(18,33)(19,31)(25,26)(28,38)(29,30)(32,37)(34,40)(36,39);(9,10,14,18)(11,16,20,19)(12,23,25,39)(13,27,29,31)(15,30,33,22)(17,35,36,24)(21,32,26,40)(28,37,38,34);(2,4)(9,11,14,20)(10,16,18,19)(12,21,25,26)(13,22,29,30)(15,31,33,27)(17,34,36,37)(23,32,39,40)(24,38,35,28);(9,12,14,25)(10,17,18,36)(11,21,20,26)(13,28,29,38)(15,32,33,40)(16,34,19,37)(22,24,30,35)(23,31,39,27)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,d^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, 404643);