Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13661 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D16 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| socle | C22 | C22×D16 |
| prime radical | C22×D16 | C22 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
386 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 386 conjugacy classes of subgroups — too many to draw. Only its 124 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)(3,6)(4,5)(7,8)(9,10,14,19)(11,15,23,30)(12,24,26,40)(13,27,25,18)(16,31,33,20)(17,34,35,21)(22,36,37,29)(28,39,32,38);(2,5)(6,8);(10,15)(11,20)(12,21)(13,25)(16,17)(18,36)(19,30)(22,39)(23,31)(24,28)(26,34)(27,29)(32,40)(33,35)(37,38);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|a^2=i,d^2=e,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fg,g^a=gh,h^a=hi,f^c=fg,g^c=gh,h^c=hi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72322380, #72322435 |
| maximal normal subgroup of | #72322380, #72322435 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420191);