Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9263 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1869 |
| prime radical | #532 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
346 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 346 conjugacy classes of subgroups — too many to draw. Only its 104 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:
(1,2,4,6,5,8,3,7)(9,10,13,18,15,19,27,33,14,11,16,21,28,22,29,35)(12,20,24,32,25,34,37,40,23,17,26,30,36,31,38,39);(2,6)(3,4)(7,8)(9,11,13,18,15,22,27,33,14,10,16,21,28,19,29,35)(12,20,26,32,25,34,38,40,23,17,24,30,36,31,37,39);(2,6)(3,4)(7,8)(9,12)(10,17)(11,20)(13,26)(14,23)(15,25)(16,24)(18,32)(19,31)(21,30)(22,34)(27,38)(28,36)(29,37)(33,40)(35,39)Polycyclic presentation:
2^9|a^2=f,b^2=df,d^2=gi,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,d^b=dg,e^b=eg,f^b=fg,d^c=dg,e^c=egi,f^c=fg86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43850);