Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143196 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1619 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#135 | D4 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
391 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 391 conjugacy classes of subgroups — too many to draw. Only its 78 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)(7,11,18,28)(8,20,23,34)(9,24,22,13)(12,30,31,21)(15,27,32,36)(16,29,19,26)(17,25,33,35);(1,2)(3,4)(5,7)(6,11)(8,15)(9,16)(10,18)(12,25)(13,26)(14,28)(17,21)(19,22)(20,27)(23,32)(24,29)(30,33)(31,35)(34,36);(1,3,4,2)(7,15)(8,21)(9,22)(11,25)(12,20)(13,24)(16,33)(17,19)(18,32)(23,30)(26,36)(27,29)(28,35)(31,34)Polycyclic presentation:
2^9|a^2=i,c^2=d,e^2=g,f^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fh,h^a=hi,c^b=cf,e^b=eh,f^b=fh,g^b=gi,h^b=hi,e^c=eg,f^c=fh,h^c=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 52187);