Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = C22×#72268043 (order 4096) · Inn(G) = C23 · Out(G) = C23×#525 |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C8 | C23 |
| commutator | C22 | C22×C16 |
| socle | C23 | C22×C8 |
| prime radical | C22×D4 | C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
289 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 289 conjugacy classes of subgroups — too many to draw. Only its 107 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:
(2,4)(5,6)(7,8)(9,10,12,16,13,17,20,23,14,11,15,18,21,19,22,24);(1,2,3,4)(9,11,12,18,13,19,20,24,14,10,15,16,21,17,22,23);(6,8)Polycyclic presentation:
2^8|a^2=f,b^2=df,f^2=g,g^2=h,b^a=bd,c^a=ce112 conjugacy classes — view the full character table.
| maximal subgroup of | #92995, #92996, #92999, … |
| maximal normal subgroup of | #92995, #92996, #92999, … |
| maximal quotient of | #136573, #136581, #136589, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5964);