Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #245246 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | #36721 | C2 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
1474 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 1474 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)(3,6)(4,7)(5,8)(9,14)(10,13)(11,15)(12,16)(20,27)(22,28)(23,30)(29,32);(1,3)(2,6)(4,9)(5,10)(7,14)(8,13)(11,16)(12,15)(17,18,21,24)(19,22,26,28)(20,23,27,30)(25,29,31,32);(1,4,5,11)(2,7,8,15)(3,9,10,16)(6,12,13,14)(17,19,21,26)(18,22,24,28)(20,25,27,31)(23,29,30,32);(17,20)(18,23)(19,25)(21,27)(22,29)(24,30)(26,31)(28,32)Polycyclic presentation:
2^9|b^2=h,c^2=i,b^a=be,c^a=cf,d^a=deh,g^a=ghi,c^b=cg,f^b=fi,e^c=eh,f^d=fh50 conjugacy classes — view the full character table.
| maximal subgroup of | #72283717, #72284555, #72284704, … |
| maximal normal subgroup of | #72283717, #72284555, #72284704, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 152455);