Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) = C22×D4×#525 (order 2048) · Inn(G) = C2×D4 · Out(G) = C24×D4 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×D4 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C4×C8 | C23 |
| socle | C22 | C2×C22⋊C8 |
| prime radical | C2×#41 | C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
180 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 73 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:
(1,2,5,8,6,3,4,7)(9,10,11,15,12,16,17,22,13,14,18,20,19,21,23,24);(1,3)(2,6)(4,8)(5,7);(1,4,6,5)(2,7,3,8)(10,14)(15,20)(16,21)(22,24)Polycyclic presentation:
2^8|a^2=e,c^2=f,d^2=f,e^2=g,g^2=h,b^a=bd,c^a=ch,d^a=df,d^b=df76 conjugacy classes — view the full character table.
| maximal subgroup of | #128636, #128637, #128652, … |
| maximal normal subgroup of | #128636, #128637, #128652, … |
| maximal quotient of | #124101, #124132, #124135, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4603);