Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9415 |
| automorphism | Aut(G) = C2×#72293988 (order 4096) · Inn(G) = #386 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | #532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
222 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 222 conjugacy classes of subgroups — too many to draw. Only its 59 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:
(1,2,5,8,6,10,4,9)(3,11,13,15,14,16,12,7)(17,18,23,26,22,19,21,25)(20,24,30,32,29,27,28,31);(1,3)(2,7,10,15)(4,12)(5,13)(6,14)(8,11,9,16)(17,19,22,18)(20,27,29,24)(21,25,23,26)(28,31,30,32);(2,8)(4,5)(7,11)(9,10)(12,13)(15,16)(17,20,22,29)(18,24,19,27)(21,30,23,28)(25,32,26,31)Polycyclic presentation:
2^8|a^2=f,b^2=dfh,c^2=h,d^2=gh,e^2=g,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dh,e^b=egh,f^b=fh,d^c=dgh,e^c=eg,f^c=fgh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9430, #200845, #200846, … |
| maximal normal subgroup of | C2×#9430, #200845, #200846, … |
| maximal quotient of | #100097, #100100, #100107, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2418);