Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #125537 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1514 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×#97 | C23 |
| prime radical | C22×D8 | C2×C4 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
311 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 311 conjugacy classes of subgroups — too many to draw. Only its 72 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)(3,5)(4,7)(6,8)(9,10,13,18,16,21,31,39,14,22,32,40,15,20,12,19)(11,23,26,37,29,33,24,35,27,38,30,17,28,34,25,36);(2,5)(3,8)(6,7)(9,11)(10,17)(12,30)(13,24)(14,27)(15,29)(16,28)(18,38)(19,34)(20,36)(21,35)(22,37)(23,40)(25,32)(26,31)(33,39);(9,12,15,32,14,31,16,13)(10,18,21,39,22,40,20,19)(11,24,28,26,27,25,29,30)(17,33,36,38,37,34,35,23)Polycyclic presentation:
2^9|a^2=e,c^2=fgi,d^2=gi,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=cfg,d^a=df,f^a=fi,c^b=ch,d^b=dg,e^b=efg,g^b=gi,h^b=hi,d^c=di,e^d=ei50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 32735);