Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9519 |
| automorphism | Aut(G) = C23×D4×#120 (order 2048) · Inn(G) = #386 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | #525 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
236 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 236 conjugacy classes of subgroups — too many to draw. Only its 61 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,6)(3,10)(4,7)(9,14)(11,12)(13,15)(17,18)(19,22)(20,27,28,23)(21,24)(25,29,31,32)(26,30);(1,2,7,14,5,9,4,6)(3,8,10,15,12,16,11,13)(17,19,21,26)(18,22,24,30)(20,25,28,31)(23,29,27,32);(1,3)(2,8)(4,11)(5,12)(6,13)(7,10)(9,16)(14,15)(17,20)(18,23)(19,25)(21,28)(22,29)(24,27)(26,31)(30,32)Polycyclic presentation:
2^8|a^2=f,b^2=dfgh,d^2=g,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=egh,e^b=eh,d^c=dh,e^c=eg,f^c=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9522, #201913, #201915, … |
| maximal normal subgroup of | C2×#9522, #201913, #201915, … |
| maximal quotient of | #100225, #100226, #100233, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2510);