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 | #36967 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1349 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 1349 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) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10)(3,7,5,9)(4,8,13,16)(11,15,12,14)(17,18)(19,23)(20,25)(21,24)(22,26)(27,30)(28,32)(29,31);(1,3,6,5)(2,7,10,9)(4,11,13,12)(8,14,16,15)(17,19)(18,23)(20,27)(21,29)(22,28)(24,31)(25,30)(26,32);(1,4)(2,8)(3,11)(5,12)(6,13)(7,14)(9,15)(10,16)(18,24)(20,21)(23,30)(25,26)(27,29)(31,32);(1,5,6,3)(2,9,10,7)(4,12,13,11)(8,15,16,14)(17,20,22,21)(18,25,26,24)(19,27,28,29)(23,30,32,31)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=hi,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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 156568);