Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #145563 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10539 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#493 | C22 |
| 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 | |
286 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 286 conjugacy classes of subgroups — too many to draw. Only its 72 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:
(1,2,6,10)(3,12)(4,7,18,21)(5,19,17,11)(8,23)(9,28,22,16)(13,27)(14,25)(15,26)(20,32)(24,31)(29,30);(2,7)(3,13)(4,16)(5,17)(8,24)(9,11)(10,21)(12,29)(14,31)(15,23)(18,28)(19,22)(20,26)(25,27);(1,3,4,14,17,23,28,32,6,15,18,31,5,13,16,30)(2,8,9,25,19,12,21,29,10,27,22,24,11,26,7,20)Polycyclic presentation:
2^9|a^2=h,c^2=d,d^2=gi,e^2=hi,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fhi,g^a=gi,c^b=cf,d^b=dg,e^b=ehi,f^b=fhi,g^b=gi,e^c=eghi,f^c=fghi,h^c=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 52759);