Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) = C24×#35680 (order 4096) · Inn(G) = C2×D4 · Out(G) = C23×#120 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C2×D4 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | C4×C4⋊C4 |
| prime radical | C23 | #117 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
171 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 171 conjugacy classes of subgroups — too many to draw. Only its 91 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,6,8,11)(7,13,14,15)(17,18)(19,22)(20,24)(21,23)(25,29)(26,28)(27,30)(31,32);(1,2)(3,7)(4,11)(5,8)(6,12)(9,15)(10,14)(13,16)(17,19,20,25,21,26,27,31)(18,22,24,29,23,28,30,32);(1,3,4,9,5,10,12,16)(2,7,11,15,8,14,6,13)(18,23)(22,28)(24,30)(29,32)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=de,d^2=g,f^2=h,b^a=bd,c^a=ch,d^b=dg,e^b=eg76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#8270, #183339, #183340, … |
| maximal normal subgroup of | C2×#8270, #183339, #183340, … |
| maximal quotient of | #96902, #96903, #96904, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1258);