Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2196 |
| 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) = C24 |
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 | |
364 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 364 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) = 12 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 12 points:
(1,2,4,3)(5,6)(7,9)(8,10)(11,12);(1,3,4,2)(6,9)(10,12);(7,11)Polycyclic presentation:
2^8|a^2=f,b^2=f,b^a=bd,c^a=ce,g^a=gh,e^b=egh,d^c=dg,e^d=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12958, #494345, #494349, … |
| maximal normal subgroup of | C2×#12958, #494345, #494349, … |
| maximal quotient of | #135952, #135966, #135970, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5946);