Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #249355 |
| 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 | #36703 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1718 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 1718 conjugacy classes of subgroups — too many to draw. Only its 152 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)(3,11,13,28)(4,15,18,24)(6,9,20,27)(7,21,23,14)(8,16,26,19)(12,25,30,29)(17,22,31,32);(4,16)(7,22)(8,24)(11,28)(14,31)(15,26)(17,21)(18,19)(23,32)(25,29);(1,3)(2,7)(4,17)(5,13)(6,12)(8,25)(9,22)(10,23)(11,15)(14,19)(16,21)(18,31)(20,30)(24,28)(26,29)(27,32);(3,12)(4,18)(7,22)(8,26)(11,29)(13,30)(14,31)(15,24)(16,19)(17,21)(23,32)(25,28)Polycyclic presentation:
2^9|a^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fhi,d^c=di,e^c=eh,f^c=fi,f^d=fh50 conjugacy classes — view the full character table.
| maximal subgroup of | #72283074, #72341155, #72341161, … |
| maximal normal subgroup of | #72283074, #72341155, #72341161, … |
| maximal quotient of | #72284697 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 156552);