Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #106 |
| commutator | C8 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C4×#106 |
| prime radical | C23 | #427 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
272 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 162 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:
(9,10,14,20,15,22,28,35,16,21,13,19,27,34,12,18)(11,23,24,33,25,36,29,32,26,39,40,38,31,37,30,17);(1,2,3,5,4,6,7,8)(9,11)(10,17,21,32)(12,24)(13,29)(14,30)(15,31)(16,26)(18,23,35,39)(19,36,20,37)(22,38,34,33)(25,27)(28,40);(9,12,27,13,16,28,15,14)(10,18,34,19,21,35,22,20)(11,24,25,29,26,40,31,30)(17,23,33,36,32,39,38,37)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=de,d^2=g,e^2=g,f^2=h,g^2=i,b^a=bd,f^a=fi,d^b=dg,e^b=egi,g^b=gi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72216215, #72216219, #72216227, … |
| maximal normal subgroup of | #72216215, #72216219, #72216227, … |
| maximal quotient of | C8×#1368, C2×#129088, #72225325, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36284);