Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2121 |
| automorphism | Aut(G) = #72281025 (order 2048) · Inn(G) = C2×#99 · Out(G) = #120 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#99 |
| commutator | C23 | C22×C8 |
| Frattini, cosocle | C2×M16 | C23 |
| socle | C2 | #1459 |
| prime radical | #525 | C4 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
184 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 184 conjugacy classes of subgroups — too many to draw. Only its 63 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 — 3 generators on 32 points:
(1,2,4,9,5,10,15,22,6,7,14,18,16,19,26,29)(3,8,12,20,13,21,25,30,11,17,23,27,24,28,31,32);(2,7)(3,11)(10,19)(12,23)(13,24)(20,27)(25,31)(30,32);(1,3,5,13,6,11,16,24)(2,8,10,21,7,17,19,28)(4,12,15,25,14,23,26,31)(9,20,22,30,18,27,29,32)Polycyclic presentation:
2^8|a^2=e,c^2=g,e^2=g,g^2=h,b^a=bd,d^a=df,f^a=fh,c^b=ch,e^b=ef,e^d=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | #129591, #129592, #129613, … |
| maximal normal subgroup of | #129591, #129592, #129613, … |
| maximal quotient of | #122360, #122361, #122364, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4472);