Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | Aut(G) = C2×#72237528 (order 2048) · Inn(G) = #120 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #120 |
| commutator | C23 | C22×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1869 |
| prime radical | #525 | C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
244 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 244 conjugacy classes of subgroups — too many to draw. Only its 81 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:
(1,2,5,10,6,11,19,25,7,3,8,13,20,14,21,27)(4,12,16,24,17,26,29,32,15,9,18,22,28,23,30,31);(1,3,5,13,6,14,19,27,7,2,8,10,20,11,21,25)(4,12,18,22,17,26,30,31,15,9,16,24,28,23,29,32);(1,4)(2,9)(3,12)(5,18)(6,17)(7,15)(8,16)(10,24)(11,23)(13,22)(14,26)(19,30)(20,28)(21,29)(25,32)(27,31)Polycyclic presentation:
2^8|a^2=f,b^2=df,f^2=g,g^2=h,b^a=bd,c^a=ce,e^a=eh,e^b=eh,d^c=dh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #129613, #129614, #129615, … |
| maximal normal subgroup of | #129613, #129614, #129615, … |
| maximal quotient of | #136576, #136584, #136592, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5969);