Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9996 |
| automorphism | Aut(G) = C23×#7737411 (order 4096) · Inn(G) = #392 · Out(G) = C26 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | #3100 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
311 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 311 conjugacy classes of subgroups — too many to draw. Only its 59 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)(4,6)(5,8)(9,10)(11,17)(12,20)(13,16)(14,19)(15,22)(18,24)(21,23);(2,6)(3,4)(7,8)(14,23)(17,24);(1,3,5,4)(2,7,8,6)(9,11)(10,14)(12,21)(13,18)(15,24)(16,23)(17,20)(19,22)Polycyclic presentation:
2^8|c^2=g,d^2=g,e^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fh,c^b=cf,d^b=dg,d^c=dh,e^c=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9997, #235215, #235216, … |
| maximal normal subgroup of | C2×#9997, #235215, #235216, … |
| maximal quotient of | #102543, #102544, #102546, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2985);