Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #112888 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #390 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1955 |
| prime radical | C22×D4 | M16 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
350 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 350 conjugacy classes of subgroups — too many to draw. Only its 92 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)(9,10,14,20,16,19,31,35,15,11,17,23,13,22,30,38)(12,24,28,39,29,21,36,37,26,25,32,40,27,18,33,34);(1,2,3,7,4,6,8,5)(9,11,14,20,16,22,31,35,15,10,17,23,13,19,30,38)(12,25,32,40,29,18,33,34,26,24,28,39,27,21,36,37);(9,12,13,27,15,26,16,29)(10,18,19,24,11,21,22,25)(14,28,30,33,17,32,31,36)(20,34,35,39,23,37,38,40)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=e,d^2=gi,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=ch,d^b=dg,e^b=ei,f^b=fg,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48055);