Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150157 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
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 | #33733 | C2 |
| 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 | |
235 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 235 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:
(1,2,4,7,5,3,8,6)(9,10,12,19,14,17,15,31)(11,21,24,38,26,16,27,32)(13,28,23,35,30,37,29,20)(18,36,33,22,34,40,39,25);(1,3)(2,5)(4,6)(7,8)(10,16)(11,22)(12,15)(13,29)(17,21)(18,35)(19,38)(20,34)(23,30)(24,36)(25,26)(27,40)(28,33)(31,32)(37,39);(2,3)(6,7)(10,17)(16,21)(18,34)(19,31)(20,35)(28,37)(32,38)(33,39)Polycyclic presentation:
2^9|a^2=e,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,c^b=cgi,d^b=dfi,e^b=ei,f^b=fh,h^b=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72216283, #72216305, #72216311, … |
| maximal normal subgroup of | #72216283, #72216305, #72216311, … |
| maximal quotient of | C2×#150160, #72321070, #72321075 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57356);