Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130207 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1786 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #62249 | C2 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
600 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 600 conjugacy classes of subgroups — too many to draw. Only its 94 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)(3,7)(6,8)(9,10,14,19,12,17,27,18,15,20,30,35,28,36,13,29)(11,21,24,33,26,37,40,31,25,38,39,16,22,32,23,34);(1,2)(3,8)(4,6)(5,7)(9,11)(10,16)(12,22)(13,24)(14,23)(15,25)(17,31)(18,37)(19,38)(20,33)(21,35)(26,28)(27,39)(29,32)(30,40)(34,36);(3,7)(5,8)(9,12,15,28)(10,17,20,36)(11,22,25,26)(13,14,27,30)(16,31,33,34)(18,35,29,19)(21,32,38,37)(23,39,40,24)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=hi,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=egi,f^b=fgh,h^b=hi,d^c=dgi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37629);