Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486811 |
| 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 | #56311 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
947 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 947 conjugacy classes of subgroups — too many to draw. Only its 116 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)(3,8,15,24)(4,7,16,9)(6,11,18,27)(12,25,13,22)(14,19,30,23)(17,20,31,26)(21,29,32,28);(2,7)(3,12)(4,16)(8,21)(9,10)(11,20)(13,15)(14,29)(17,31)(19,25)(22,23)(24,32)(26,27)(28,30);(3,13)(4,17)(8,22)(12,15)(14,30)(16,31)(19,32)(21,23)(24,25)(28,29);(1,3,6,12)(2,8,11,25)(4,14,17,28)(5,15,18,13)(7,19,20,21)(9,23,26,32)(10,24,27,22)(16,30,31,29)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,d^b=di,e^b=eg,f^b=fhi,d^c=dgi,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, 394139);