Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | Aut(G) = C2×#72231730 (order 4096) · Inn(G) = C2×D8 · Out(G) = C22×#120 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | C2×D8 |
| commutator | C8 | C23×C4 |
| Frattini, cosocle | C2×C8 | C24 |
| socle | C2 | C2×C4×D8 |
| prime radical | #3414 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
326 conjugacy classes of subgroups; only the 157 normal subgroups are shown.
This group has 326 conjugacy classes of subgroups — too many to draw. Only its 157 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,10,22,12)(3,13,15,24,17,30,16,32)(4,9,18,28,19,27,5,20)(8,23,26,31,25,21,29,14);(1,3)(2,8)(4,14)(5,21)(6,15)(7,17)(9,24)(10,25)(11,26)(12,29)(13,20)(16,22)(18,23)(19,31)(27,32)(28,30);(1,4,6,18,7,19,22,5)(2,9,11,28,10,27,12,20)(3,14,15,23,17,31,16,21)(8,24,26,30,25,32,29,13);(2,10)(8,25)(9,27)(11,12)(13,30)(20,28)(24,32)(26,29)Polycyclic presentation:
2^8|a^2=f,c^2=f,e^2=gh,f^2=h,g^2=h,b^a=be,d^a=dh,e^a=eg,g^a=gh,e^b=eg,g^b=gh64 conjugacy classes — view the full character table.
| maximal subgroup of | #136679, #136680, #136681, … |
| maximal normal subgroup of | #136679, #136680, #136681, … |
| maximal quotient of | C8×#509, C8×#510, #483784, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26451);