Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1304 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #95 |
| commutator | C23 | C4×C8 |
| Frattini, cosocle | C24×C4 | C22 |
| prime radical | C25 | C2×C4 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C23 ▷ C1 | |
365 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 365 conjugacy classes of subgroups — too many to draw. Only its 67 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 — 2 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10)(11,15)(12,14)(13,16)(17,18,19,21,20,22,23,24);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13,14,15)(12,16)Polycyclic presentation:
2^8|a^2=d,b^2=e,d^2=h,b^a=bc,c^a=cf,e^a=eg,c^b=cg,d^b=df52 conjugacy classes — view the full character table.
| maximal subgroup of | #93901, #93943, #93965, … |
| maximal normal subgroup of | #93901, #93943, #93965, … |
| maximal quotient of | #92807, #92808, #92812, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20);