Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130284 |
| 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 | #62269 | 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 | |
620 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 620 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:
(1,2,4,9,14,24,16,30,5,11,18,28,6,10,19,27)(3,8,15,25,29,21,31,32,13,23,17,26,12,7,20,22);(2,7)(3,12)(4,16)(6,14)(8,10)(9,26)(11,21)(13,29)(18,19)(20,31)(22,27)(23,24)(25,28)(30,32);(3,13)(12,29)(15,17)(20,31)Polycyclic presentation:
2^9|a^2=f,d^2=hi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,e^b=egi,f^b=fgh,h^b=hi,d^c=dg,e^d=ei,f^d=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72231599, #72231603, #72231615, … |
| maximal normal subgroup of | #72231599, #72231603, #72231615, … |
| maximal quotient of | #72250401, #72250687, C2×#130288, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37484);