Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #487643 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | #58260 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1036 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 1036 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:
(1,2,5,10)(3,12,17,8)(4,7,18,9)(6,11,15,24)(13,23,14,26)(16,25,32,20)(19,21,31,27)(22,29,28,30);(2,7)(3,13)(4,18)(8,22)(9,10)(11,21)(12,28)(14,17)(16,30)(19,31)(20,26)(23,25)(24,27)(29,32);(3,14)(4,19)(8,23)(12,26)(13,17)(16,32)(18,31)(20,28)(22,25)(29,30);(1,3,15,14)(2,8,24,23)(4,16,31,30)(5,17,6,13)(7,20,27,28)(9,25,21,22)(10,12,11,26)(18,32,19,29)Polycyclic presentation:
2^9|a^2=g,d^2=gi,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,d^b=di,e^b=eg,f^b=fhi,d^c=dgi,e^c=eh,f^e=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 395187);