Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486819 |
| 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×C22⋊C4 | C24 |
| prime radical | #52309 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
992 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 992 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)(3,8,13,22)(4,7,15,9)(6,11,18,26)(12,24,27,32)(14,19,28,23)(16,21,29,31)(17,20,30,25);(2,7)(3,12)(4,15)(8,21)(9,10)(11,20)(13,27)(14,29)(16,28)(17,30)(19,24)(22,31)(23,32)(25,26);(1,3,6,12)(2,8,11,24)(4,16,17,14)(5,13,18,27)(7,19,20,21)(9,23,25,31)(10,22,26,32)(15,29,30,28);(3,13)(8,22)(12,27)(14,28)(16,29)(19,23)(21,31)(24,32)Polycyclic presentation:
2^9|a^2=g,c^2=i,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,c^b=ci,e^b=eg,f^b=fhi,d^c=dg,e^c=eh,f^e=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 394145);