Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #141751 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1957 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #492 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
392 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 392 conjugacy classes of subgroups — too many to draw. Only its 122 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:
(2,5)(3,7)(6,8)(9,10,13,18,15,19,14,30)(11,20,22,37,23,36,29,38)(12,24,26,32,28,40,27,17)(16,31,33,21,35,39,34,25);(2,6)(5,8)(9,11,12,25,14,22,27,31,15,23,28,21,13,29,26,39)(10,16,32,20,18,34,40,38,19,35,17,36,30,33,24,37);(1,2,3,5,4,6,7,8)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=e,d^2=g,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,c^b=ch,d^b=di,f^b=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72225232, #72225234, #72226939, … |
| maximal normal subgroup of | #72225232, #72225234, #72226939, … |
| maximal quotient of | #72226939 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48949);