Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1665 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #121 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #389 |
| prime radical | C23 | C22⋊Q8 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
161 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 161 conjugacy classes of subgroups — too many to draw. Only its 77 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)(3,5)(4,7)(6,8)(9,10,14,18,15,19,13,23)(11,16,20,24,21,17,12,22);(2,5)(3,8)(6,7)(9,11,14,20,15,21,13,12)(10,16,18,24,19,17,23,22);(9,12,13,21,15,20,14,11)(10,17,18,22,19,16,23,24)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=e,d^2=g,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dg58 conjugacy classes — view the full character table.
| maximal subgroup of | #136341, #136386, #136387, … |
| maximal normal subgroup of | #136341, #136386, #136387, … |
| maximal quotient of | #101927, #101930, #101945, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2615);