Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1339 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #327 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×#97 | C23 |
| socle | C22 | C2×#327 |
| prime radical | C24 | #102 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
401 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 401 conjugacy classes of subgroups — too many to draw. Only its 102 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:
(1,2,4,7,5,8,3,6)(9,10,13,18,15,19,14,30)(11,20,22,34,24,37,23,40)(12,25,27,17,29,35,28,36)(16,26,31,39,33,38,32,21);(9,11,15,24)(10,16,19,33)(12,26,29,38)(13,23,14,22)(17,34,36,40)(18,31,30,32)(20,35,37,25)(21,28,39,27);(1,3,5,4)(2,6,8,7)Polycyclic presentation:
2^9|a^2=e,b^2=i,c^2=hi,d^2=gi,e^2=h,f^2=i,g^2=i,b^a=bd,d^a=df,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31361);