Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #251827 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | #46996 | C2 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
1266 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 1266 conjugacy classes of subgroups — too many to draw. Only its 130 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 — 4 generators on 32 points:
(2,7)(4,14)(8,22)(9,19)(10,21)(13,29)(15,16)(18,23)(20,30)(24,31)(25,32)(27,28);(2,8,9,23)(3,11)(4,15)(7,18,19,22)(10,24,25,30)(12,26)(13,29)(14,16)(20,32,31,21)(27,28);(1,2,6,10)(3,7,11,21)(4,8,16,24)(5,9,17,25)(12,19,26,32)(13,20,27,18)(14,30,15,23)(22,28,31,29);(3,12)(7,19)(11,26)(13,28)(18,22)(20,31)(21,32)(27,29)Polycyclic presentation:
2^9|b^2=e,c^2=i,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,f^b=fi,g^b=gh,e^c=eh,f^d=fh50 conjugacy classes — view the full character table.
| maximal subgroup of | #72234597, #72234626, #72234627, … |
| maximal normal subgroup of | #72234597, #72234626, #72234627, … |
| maximal quotient of | #72251578, #72251588, #72251608, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 159025);