Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144993 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10408 |
| 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 | |
403 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 403 conjugacy classes of subgroups — too many to draw. Only its 76 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:
(2,7)(3,12)(4,17)(5,18)(9,11)(10,26)(13,30)(15,20)(21,32)(22,23)(27,28)(29,31);(1,2)(3,13)(4,9)(5,19)(6,11)(7,17)(8,15)(10,18)(12,28)(14,30)(16,27)(20,23)(21,25)(22,31)(24,29)(26,32);(1,3,4,14,6,12,17,16)(2,8,7,22,11,24,9,23)(5,20,18,29,21,31,32,15)(10,27,19,28,26,30,25,13)Polycyclic presentation:
2^9|c^2=d,d^2=i,e^2=g,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fh,h^a=hi,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=gi,h^b=hi,e^c=egi,f^c=fhi,h^c=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 52189);