Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94648 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7403 |
| commutator | C2×#97 | C2×C4 |
| Frattini, cosocle | #1961 | C22 |
| prime radical | C2×#127 | D4 |
| LCS | G ▷ C2×#97 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ #1961 ◁ G | |
| LECS | G ▷ #1961 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
184 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 184 conjugacy classes of subgroups — too many to draw. Only its 24 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)(3,11,13,28)(4,16,18,8)(5,10)(6,9,20,27)(7,12,21,31)(14,29,15,24)(17,23)(19,25,30,26)(22,32);(1,3,5,14)(2,7)(4,17,19,32)(6,13,20,15)(8,24)(9,23)(10,21)(11,25)(12,30,31,18)(16,29)(22,27)(26,28)Polycyclic presentation:
2^9|a^2=d,b^2=h,c^2=f,e^2=hi,f^2=i,g^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fh,g^a=gh,h^a=hi,c^b=cf,d^b=defh,e^b=ehi,f^b=fi,d^c=dgh,e^c=ei,e^d=ehi,f^d=fi,g^d=gi29 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1844);