Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8510 |
| automorphism | Aut(G) = C23×D4×#120 (order 2048) · Inn(G) = #379 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #379 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | #3561 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
364 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 364 conjugacy classes of subgroups — too many to draw. Only its 73 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:
(2,5,6,7)(10,15)(11,18)(12,13)(17,20)(19,21)(22,23);(1,2)(3,7)(4,6)(5,8)(9,10)(11,16)(12,20)(13,15)(14,17)(18,23)(19,24)(21,22);(3,8)(5,7)(9,11,14,19)(10,16,17,24)(12,18,13,21)(15,22,20,23)Polycyclic presentation:
2^8|a^2=f,c^2=g,d^2=h,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=egh,f^b=fgh,d^c=dgh,e^c=eg40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#8511, #187660, #187664, … |
| maximal normal subgroup of | C2×#8511, #187660, #187664, … |
| maximal quotient of | #97009, #97010, #97013, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1499);