Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12767 |
| automorphism | Aut(G) = #72256403 (order 2048) · Inn(G) = #1790 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1790 |
| commutator | C2×D4 | C22×C4 |
| Frattini, cosocle | C22×D4 | C23 |
| prime radical | #3543 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
439 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 439 conjugacy classes of subgroups — too many to draw. Only its 69 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)(3,10,12,7)(4,8,14,15)(5,9)(6,11)(13,16);(1,3)(2,6)(4,11)(5,12)(7,15)(8,10)(9,16)(13,14);(1,3)(2,7)(4,13)(5,12)(6,15)(8,16)(9,10)(11,14)Polycyclic presentation:
2^8|a^2=f,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,e^b=egh,f^b=fg,d^c=dg,f^c=fh,e^d=eh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #152572, #152598, #152599, … |
| maximal normal subgroup of | #152572, #152598, #152599, … |
| maximal quotient of | #129575, #129577, #129579, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5755);