Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13096 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1898 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×#117 | C23 |
| prime radical | C22×AGL(1,ℤ/8) | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#520 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
666 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 666 conjugacy classes of subgroups — too many to draw. Only its 104 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) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,7,5,6,8,3)(9,10,14,19)(11,20)(12,26)(13,18,30,35)(15,22,33,23)(16,29)(17,21)(24,37)(25,38)(27,34)(28,31)(32,39,36,40);(1,3,4,6,5,7,8,2)(9,11,21,23,14,25,12,22)(10,15,26,38,19,33,17,20)(13,28,29,39,30,24,27,40)(16,31,18,36,34,37,35,32);(9,12,14,21)(10,16,19,34)(11,22,25,23)(13,29,30,27)(15,31,33,37)(17,35,26,18)(20,32,38,36)(24,40,28,39)Polycyclic presentation:
2^9|a^2=f,b^2=dfi,c^2=i,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dg,e^b=ei,f^b=fg,g^b=gi,d^c=di,e^c=ei,f^c=fi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44883);