Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #346224 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2760 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | #10336 | C2 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
743 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 743 conjugacy classes of subgroups — too many to draw. Only its 92 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)(3,9,11,6)(4,12,13,7)(5,8)(10,15)(14,16)(17,18)(19,22,26,29)(20,27,28,23)(21,24)(25,31)(30,32);(1,3,5,11)(2,6,8,9)(4,10,13,16)(7,15,12,14)(22,29)(25,32);(1,4,5,13)(2,7,8,12)(3,10)(6,14)(9,15)(11,16)(17,19)(18,22,24,29)(20,25)(21,26)(23,30,27,31)(28,32);(6,9)(14,15)(17,20)(18,23)(19,25)(21,28)(22,30)(24,27)(26,32)(29,31)Polycyclic presentation:
2^9|a^2=fgh,b^2=h,c^2=efh,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fhi,g^a=ghi,c^b=cg,d^b=de,f^b=fh,e^c=ei,f^c=fi,g^c=gh,g^d=gi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72283479, #72283535, #72283537, … |
| maximal normal subgroup of | #72283479, #72283535, #72283537, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 253427);