Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153123 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2196 |
| commutator, Frattini, cosocle | C2×C4⋊Q8 | C23 |
| prime radical | C2×2−1+4 | D4 |
| LCS | G ▷ C2×C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊Q8 ◁ G | |
| LECS | G ▷ C2×C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
405 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 405 conjugacy classes of subgroups — too many to draw. Only its 48 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,3)(5,6,10,14)(7,11,19,30)(8,20,21,32)(9,23,25,13)(12,24,31,22)(15,33,17,36)(16,29,18,27)(26,35,28,34);(1,2)(3,4)(5,7)(6,11)(8,15)(9,16)(10,19)(12,26)(13,27)(14,30)(17,21)(18,25)(20,33)(22,34)(23,29)(24,35)(28,31)(32,36);(1,3,4,2)(7,15)(8,21)(11,26)(12,31)(16,35)(17,19)(18,34)(20,32)(22,24)(27,36)(28,30)(29,33)Polycyclic presentation:
2^9|a^2=i,c^2=d,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fgh,g^a=gi,c^b=cf,e^b=eg,f^b=fi,g^b=gi,h^b=hi,e^c=eh,f^c=fi,g^c=gh,f^e=fh,g^f=gi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60484);