Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #487707 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C4⋊C4 | C24 |
| prime radical | #53422 | C2 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1006 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 1006 conjugacy classes of subgroups — too many to draw. Only its 118 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 — 4 generators on 32 points:
(1,2,6,13)(3,16,21,10)(4,8,22,11)(5,9,25,32)(7,15,26,14)(12,24,27,19)(17,31,18,28)(20,29,23,30);(2,8)(3,17)(4,22)(5,24)(9,16)(10,32)(11,13)(12,31)(14,29)(15,30)(18,21)(19,25)(20,23)(27,28);(2,9)(3,18)(4,22)(5,19)(8,16)(10,11)(12,15)(13,32)(14,27)(17,21)(20,23)(24,25)(28,29)(30,31);(1,3,6,21)(2,10,13,16)(4,19,22,24)(5,20,25,23)(7,17,26,18)(8,27,11,12)(9,30,32,29)(14,31,15,28)Polycyclic presentation:
2^9|a^2=g,d^2=g,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,h^a=hi,d^b=di,e^b=eg,f^b=fhi,d^c=dgi,e^c=eh,f^c=fi,f^e=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 395223);