Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2199 |
| automorphism | Aut(G) = C25×B(4,𝔽2) (order 2048) · Inn(G) = B(4,𝔽2) · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | B(4,𝔽2) |
| commutator | C2×D4 | C22×C4 |
| Frattini, cosocle | C22×D4 | C23 |
| prime radical | C2×#120 | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
304 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 304 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) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,4,3)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20);(1,3,4,2)(5,7)(6,10)(8,15)(9,14)(11,13)(12,17)(16,20)(18,19);(7,14)(8,16)(10,17)(13,19)Polycyclic presentation:
2^8|a^2=f,b^2=f,b^a=bd,c^a=ce,g^a=gh,c^b=ch,e^b=egh,d^c=dg,e^d=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12960, #494348, #494349, … |
| maximal normal subgroup of | C2×#12960, #494348, #494349, … |
| maximal quotient of | #135954, #135968, #135972, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5948);