Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150138 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#357 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C4×C16 | C23 |
| prime radical | C23 | #357 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
240 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 240 conjugacy classes of subgroups — too many to draw. Only its 78 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 — 3 generators on 40 points:
(1,2,3,6,4,5,7,8)(10,15,16,30)(11,19)(12,24)(13,26)(17,34,33,39)(18,35,28,32)(20,25,31,29)(21,27)(22,40)(23,38)(36,37);(9,10,14,16)(11,20,23,31)(12,25,27,29)(13,28,26,18)(15,19,30,38)(17,21,33,24)(22,34,36,39)(32,40,35,37);(2,5)(6,8)Polycyclic presentation:
2^9|a^2=e,b^2=i,d^2=fhi,e^2=g,f^2=hi,h^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,d^b=dfi,e^b=ei,f^b=fh,h^b=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72234339, #72234342, #72234353, … |
| maximal normal subgroup of | #72234339, #72234342, #72234353, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57335);