Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94791 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7493 |
| commutator | #127 | C42 |
| Frattini, prime radical, cosocle | #3151 | C22 |
| LCS | G ▷ #127 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ #3151 ◁ G | |
| LECS | G ▷ #3151 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
212 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 34 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 — 2 generators on 32 points:
(1,2,4,9,6,11,18,27)(3,12,14,29,13,23,16,26)(5,8,17,25,20,10,19,28)(7,21,22,30,24,31,32,15);(1,3)(2,7)(4,16,18,14)(5,15)(6,13)(8,12)(9,26,27,29)(10,23)(11,24)(17,31,19,21)(20,30)(22,28,32,25)Polycyclic presentation:
2^9|a^2=d,b^2=e,d^2=i,f^2=hi,g^2=i,b^a=bc,c^a=cf,e^a=eh,f^a=fg,g^a=gh,h^a=hi,c^b=ch,d^b=dfhi,f^b=fh,g^b=ghi,d^c=dgh,f^c=fhi,g^c=gi,e^d=ei,f^d=fhi,g^d=gi29 conjugacy classes — view the full character table.
| maximal subgroup of | #72245221, #72245231, #72245347, … |
| maximal normal subgroup of | #72245221, #72245231, #72245347, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1989);