Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486133 |
| 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×C4⋊C4 | C24 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
950 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 950 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)(3,16)(4,12)(5,9)(6,24)(7,13)(8,15)(10,18)(11,22)(14,26)(17,27)(19,28)(20,32)(21,31)(23,30)(25,29);(2,9)(3,17)(4,20)(5,18)(6,22)(10,13)(11,28)(12,16)(14,29)(15,30)(19,21)(23,25)(24,27)(31,32);(1,3,8,17)(2,10,15,29)(4,18,6,25)(5,20,23,22)(7,19,26,21)(9,14,30,13)(11,31,32,28)(12,27,24,16);(1,4,8,6)(2,11,15,32)(3,18,17,25)(5,21,23,19)(7,22,26,20)(9,27,30,16)(10,31,29,28)(12,14,24,13)Polycyclic presentation:
2^9|c^2=i,d^2=i,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,h^a=hi,c^b=ci,d^b=dgi,e^b=eg,f^b=fhi,e^c=eh,f^c=fi,f^e=fi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 393332);