Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #140370 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1943 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #441 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1450 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
208 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 208 conjugacy classes of subgroups — too many to draw. Only its 90 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:
(1,2,6,3,5,7,4,8)(10,16)(11,20,23,36)(12,14)(13,27)(17,31,33,34)(18,28)(19,25)(21,32,38,40)(22,37,24,35)(26,29);(1,3,6,7,5,8,4,2)(9,10)(11,21,23,38)(12,25)(13,28)(14,16)(15,19)(17,24,33,22)(18,29)(20,34,36,31)(26,30)(27,39)(32,35,40,37);(1,4,5,6)(2,3,7,8)(9,11,13,20,14,24,27,37,15,23,29,36,12,22,26,35)(10,17,18,32,16,21,30,31,19,33,28,40,25,38,39,34)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=e,d^2=i,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gi,c^b=cgh,d^b=di,e^b=ei,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72321061, #72321077, #72321111 |
| maximal normal subgroup of | #72321061, #72321077, #72321111 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47566);