Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130292 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #8350 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | C2×#131 | C23 |
| prime radical | #62283 | C2 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
622 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 622 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,10,5,11,17,27,6,8,18,23,15,24,30,32)(3,12,16,26,13,21,29,31,14,9,19,25,28,7,20,22);(2,7)(3,13)(4,17)(5,15)(8,21)(9,11)(10,25)(12,24)(14,28)(16,19)(18,30)(22,32)(23,26)(27,31);(2,8)(3,14)(4,18)(7,21)(9,12)(11,24)(13,28)(17,30)Polycyclic presentation:
2^9|a^2=f,d^2=hi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dh,e^b=eg,f^b=fgh,h^b=hi,d^c=dgi,f^c=fi,e^d=ei,f^d=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72231596, #72231603, #72231624, … |
| maximal normal subgroup of | #72231596, #72231603, #72231624, … |
| maximal quotient of | #72240706, #72240716, #72250393, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37488);