Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C2×D8 |
| commutator | C8 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C4×#107 |
| prime radical | C23 | C4×Q16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
254 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 254 conjugacy classes of subgroups — too many to draw. Only its 142 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:
(1,2,4,7,5,3,6,8)(9,10,12,18,14,20,28,36,15,17,29,33,13,19,27,35)(11,21,22,31,23,38,39,30,25,37,40,32,24,34,26,16);(9,11,13,23,15,25,14,24)(10,16,19,31,17,30,20,32)(12,26,27,22,29,39,28,40)(18,34,35,21,33,38,36,37);(1,3,4,8,5,2,6,7)(10,17)(16,30)(18,33)(19,20)(21,37)(31,32)(34,38)(35,36)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=deg,d^2=gi,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ci,d^b=dg,e^b=eg,g^b=gi,h^b=hi128 conjugacy classes — view the full character table.
| maximal subgroup of | C8×#2179, #72223674, #72223677, … |
| maximal normal subgroup of | C8×#2179, #72223674, #72223677, … |
| maximal quotient of | C2×#129149 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36345);