Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130848 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #8376 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×#131 | C23 |
| prime radical | #61267 | C2 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
579 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 579 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:
(1,2,5,12,6,13,22,23,7,11,14,28,20,27,30,32)(3,9,16,10,17,26,31,8,15,25,4,18,21,29,19,24);(2,8)(4,19)(5,22)(6,20)(9,23)(10,13)(11,24)(12,26)(14,30)(16,31)(17,21)(18,27)(25,32)(28,29);(1,3)(2,9,11,25)(4,14)(5,16)(6,17)(7,15)(8,23,24,32)(10,12,18,28)(13,26,27,29)(19,30)(20,21)(22,31)Polycyclic presentation:
2^9|a^2=f,c^2=g,d^2=hi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,f^b=fgh,h^b=hi,e^c=ei,f^c=fg,f^d=fi,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72231596, #72231701, #72231719, … |
| maximal normal subgroup of | #72231596, #72231701, #72231719, … |
| maximal quotient of | #72250277, #72250881, #72280189, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 38044);