Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11796 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×#97 | C23 |
| socle | C22 | #1459 |
| prime radical | C24 | C22⋊C8 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
307 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 307 conjugacy classes of subgroups — too many to draw. Only its 80 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:
(1,2,5,7,6,3,4,8)(9,10,13,19,14,18,27,33,15,20,26,34,12,16,25,29)(11,17,22,31,23,30,37,39,24,32,36,40,21,28,35,38);(1,3)(2,6)(4,8)(5,7)(10,16,20,18)(13,26)(17,28,32,30)(19,29,34,33)(22,36)(25,27)(31,38,40,39)(35,37);(1,4,6,5)(2,7,3,8)(9,11,14,23,15,24,12,21)(10,17,18,30,20,32,16,28)(13,22,27,37,26,36,25,35)(19,31,33,39,34,40,29,38)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=h,d^2=gi,e^2=h,h^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fi,d^b=dg,e^b=efgi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31810);