Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1879 |
| prime radical | C22×D16 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ #1450 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
404 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 404 conjugacy classes of subgroups — too many to draw. Only its 106 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 — 3 generators on 40 points:
(1,2,4,7,5,3,6,8)(9,10,12,18,15,21,13,31,16,22,30,38,14,20,29,19)(11,23,24,36,27,34,40,37,28,39,32,35,26,33,25,17);(1,3,4,8,5,2,6,7)(9,11)(10,17)(12,25)(13,32)(14,27)(15,26)(16,28)(18,33)(19,23)(20,36)(21,35)(22,37)(24,29)(30,40)(31,39)(34,38);(10,18)(12,29)(13,30)(14,15)(17,33)(19,21)(20,31)(22,38)(23,35)(24,25)(26,27)(32,40)(34,37)(36,39)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=gi,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=ei,d^b=dg,e^b=eg,f^b=fg,g^b=gi,h^b=hi,d^c=dg,e^c=egi,f^c=fg,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43885);