Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13307 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1874 |
| prime radical | C2×#41 | C22⋊C4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
398 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 398 conjugacy classes of subgroups — too many to draw. Only its 102 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:
(2,6,7,8)(9,10,14,21,13,20,30,17,16,23,12,19,15,22,31,33)(11,24,28,32,25,18,34,35,29,37,26,39,27,38,40,36);(1,2,3,6,5,7,4,8)(10,17)(11,25)(12,30)(13,15)(14,31)(18,35)(19,22)(20,21)(23,33)(24,39)(27,29)(32,37)(34,40)(36,38);(1,3,5,4)(2,6,7,8)(9,11,15,27,16,29,13,25)(10,18,22,24,23,38,20,37)(12,28,30,40,14,26,31,34)(17,32,21,36,33,39,19,35)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=g,d^2=gi,e^2=i,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,c^b=ch,d^b=dg,e^b=ei,f^b=fg,g^b=gi,h^b=hi,d^c=di,e^c=ei,f^c=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43832);