Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) = C23×D4×#392 (order 4096) · Inn(G) = C2×D4 · Out(G) = C22×#392 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C2×D4 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | C4×C4⋊C4 |
| prime radical | C23 | C4×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
171 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 171 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,3,5,6,8,7)(10,15,17,21)(16,22,23,24);(9,10,13,15,14,17,12,21)(11,16,19,22,20,23,18,24);(1,3,4,6,5,7,8,2)(9,11,12,18,14,20,13,19)(10,16,21,24,17,23,15,22)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=de,d^2=g,e^2=h,f^2=g,b^a=bd,c^a=ch,d^b=dg,e^b=eg76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#8296, #183506, #183507, … |
| maximal normal subgroup of | C2×#8296, #183506, #183507, … |
| maximal quotient of | #96896, #96897, #96902, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1284);