Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13595 |
| automorphism | Aut(G) = #72319052 (order 4096) · Inn(G) = C2×#354 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#354 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | #127 | C23 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×#127 ◁ G | |
| LECS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
194 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 194 conjugacy classes of subgroups — too many to draw. Only its 45 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,10,12,15)(4,8,14,7)(5,9)(6,11,16,13);(1,3,5,11)(2,6)(4,13,14,12)(7,10)(8,16)(9,15);(3,11)(6,15)(10,16)(12,13)Polycyclic presentation:
2^8|a^2=e,b^2=g,d^2=gh,f^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,c^b=ch,d^b=dg,e^b=efg,f^b=fh,e^d=egh,f^e=fh28 conjugacy classes — view the full character table.
| maximal subgroup of | #94772, #152069, #152085, … |
| maximal normal subgroup of | #94772, #152069, #152085, … |
| maximal quotient of | #149583, #149615, #149703, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6583);