Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | Aut(G) = #72303323 (order 2048) · Inn(G) = C2×C22⋊C4 · Out(G) = C22×#41 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×C22⋊C4 |
| commutator | C23 | C22×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C2×C22⋊C8 |
| prime radical | C22×D4 | C8 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
209 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 209 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) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10,12,17,14,18,21,24,13,15,11,16,19,22,20,23);(1,3)(2,5)(4,8)(6,7)(10,15)(11,12)(18,22)(20,21);(9,11,14,20,13,12,19,21)(10,16,18,23,15,17,22,24)Polycyclic presentation:
2^8|a^2=e,c^2=g,e^2=g,g^2=h,b^a=bd,d^a=df,c^b=ch,e^b=ef64 conjugacy classes — view the full character table.
| maximal subgroup of | #129571, #129572, #129597, … |
| maximal normal subgroup of | #129571, #129572, #129597, … |
| maximal quotient of | #122366, #122367, #122368, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4455);