Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | Aut(G) = C22×#72219693 (order 4096) · Inn(G) = C2×C22⋊C4 · Out(G) = C24×D4 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | C4×C22⋊C4 |
| prime radical | #532 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
222 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 222 conjugacy classes of subgroups — too many to draw. Only its 85 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:
(2,5)(3,7)(9,10,13,17,14,18,12,15)(11,16,20,23,21,24,19,22);(1,2,3,6,4,5,7,8)(10,15,18,17)(16,22,24,23);(2,5)(6,8)(9,11,13,20,14,21,12,19)(10,16,17,23,18,24,15,22)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=e,d^2=g,e^2=g,f^2=h,b^a=bd,d^a=dg,f^a=fgh,c^b=ch,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #140165, C2×#8245, #182881, … |
| maximal normal subgroup of | #140165, C2×#8245, #182881, … |
| maximal quotient of | #96764, #96766, #96771, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1233);