Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #249319 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1715 conjugacy classes of subgroups; only the 156 normal subgroups are shown.
This group has 1715 conjugacy classes of subgroups — too many to draw. Only its 156 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:
(1,2,5,10)(3,11,12,28)(4,15,18,24)(6,9,20,27)(7,21,23,13)(8,16,26,19)(14,25,29,32)(17,22,30,31);(4,16)(7,22)(8,24)(11,28)(13,30)(15,26)(17,21)(18,19)(23,31)(25,32);(1,3,6,14)(2,7,9,22)(4,17,19,13)(5,12,20,29)(8,25,15,11)(10,23,27,31)(16,21,18,30)(24,28,26,32);(4,18)(8,26)(11,28)(13,21)(15,24)(16,19)(17,30)(25,32)Polycyclic presentation:
2^9|a^2=h,c^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fhi,e^c=eh,f^c=fi,f^d=fh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72227688, #72282959, #72282999, … |
| maximal normal subgroup of | #72227688, #72282959, #72282999, … |
| maximal quotient of | #72209853, #72284235 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 156528);