Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13309 |
| automorphism | Aut(G) = C2×#72231699 (order 2048) · Inn(G) = #469 · Out(G) = C25 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #469 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #2065 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
157 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 157 conjugacy classes of subgroups — too many to draw. Only its 53 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 — 3 generators on 32 points:
(1,2,9,14)(3,15)(4,21)(5,12,26,32)(6,11,7,29)(8,16,28,13)(10,23)(17,25)(18,31)(19,22)(20,30)(24,27);(1,3,5,17,8,19,6,27,9,20,26,21,28,10,7,18)(2,4,11,15,13,24,12,23,14,25,29,30,16,31,32,22);(1,4)(2,10)(3,16)(5,23)(6,15)(7,30)(8,31)(9,25)(11,17)(12,18)(13,20)(14,19)(21,29)(22,26)(24,28)(27,32)Polycyclic presentation:
2^8|a^2=de,b^2=d,d^2=g,e^2=gh,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fg,g^a=gh,c^b=cf,e^b=eh,d^c=dg,e^c=eg,f^c=fg,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13466, #499302, #499478, … |
| maximal normal subgroup of | C2×#13466, #499302, #499478, … |
| maximal quotient of | #145908, #145918, #145950, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6454);