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) = C23×C4 |
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 | C24 | C2×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
548 conjugacy classes of subgroups; only the 210 normal subgroups are shown.
This group has 548 conjugacy classes of subgroups — too many to draw. Only its 210 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 44 points:
(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,14,16,22,28,34,40,30,17,20,29,32,18,23,25,19)(15,21,24,35,37,44,41,42,26,33,38,43,27,36,39,31);(5,7,8,11)(6,9,10,12)(14,19)(15,24)(16,25)(18,28)(20,30)(22,23)(26,38)(27,41)(29,40)(31,35)(32,34)(36,44)(37,39)(42,43);(6,10)(9,12)(14,20)(19,30)(21,33)(22,32)(23,34)(31,42)(35,43)(36,44);(13,15,25,39,18,27,29,38,17,26,40,41,28,37,16,24)(14,21,19,31,23,36,32,43,20,33,30,42,34,44,22,35)Polycyclic presentation:
2^9|a^2=g,b^2=fi,d^2=eh,e^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^b=de,e^b=eh,g^b=gh,h^b=hi110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404965);