Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11753 |
| 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 | |
311 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 311 conjugacy classes of subgroups — too many to draw. Only its 82 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:
(2,5,6,7)(9,10,13,19,14,18,29,34,15,20,27,36,12,16,26,31)(11,21,24,35,25,30,38,39,22,17,28,33,23,32,37,40);(1,2)(3,7)(4,6)(5,8)(10,16,20,18)(11,22)(13,27)(17,32,21,30)(19,31,36,34)(23,25)(26,29)(33,40,35,39);(9,11,14,25,15,22,12,23)(10,17,18,32,20,21,16,30)(13,28,29,37,27,24,26,38)(19,35,34,39,36,33,31,40)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=cf,d^a=df,f^a=fi,c^b=ci,d^b=dg,e^b=efgi,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31600);