Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143322 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10054 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #28504 | C2 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
488 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 488 conjugacy classes of subgroups — too many to draw. Only its 80 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 — 3 generators on 32 points:
(1,2)(3,11,14,30)(4,16)(5,21)(6,9)(7,13,23,31)(8,19)(10,22)(12,29,15,27)(17,28)(18,24,32,25)(20,26);(4,17)(5,22)(7,23)(8,26)(10,21)(11,29)(12,15)(13,18)(16,28)(19,20)(27,30)(31,32);(1,3)(2,7)(4,18)(5,12)(6,14)(8,27)(9,23)(10,24)(11,26)(13,17)(15,22)(16,30)(19,31)(20,32)(21,25)(28,29)Polycyclic presentation:
2^9|a^2=g,d^2=g,e^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=dg,e^b=ehi,h^b=hi,d^c=dh,e^c=ehi,g^c=gi,h^c=hi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72214311, #72214383, #72214424, … |
| maximal normal subgroup of | #72214311, #72214383, #72214424, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50518);