Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2199 |
| automorphism | Aut(G) = #72234841 (order 2048) · Inn(G) = B(4,𝔽2) · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | B(4,𝔽2) |
| commutator | C2×D4 | C22×C4 |
| Frattini, cosocle | C2×#41 | C23 |
| socle | C2 | #1867 |
| prime radical | #532 | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
208 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 208 conjugacy classes of subgroups — too many to draw. Only its 57 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:
(1,2,6,11,7,12,5,20)(3,13,16,27,17,30,21,24)(4,10,18,9,19,28,22,29)(8,23,25,14,26,31,32,15);(1,3,6,16,7,17,5,21)(2,8,11,25,12,26,20,32)(4,15,18,23,19,14,22,31)(9,27,28,30,29,24,10,13);(1,4)(2,9)(3,14)(5,22)(6,18)(7,19)(8,24)(10,20)(11,28)(12,29)(13,25)(15,17)(16,31)(21,23)(26,27)(30,32)Polycyclic presentation:
2^8|a^2=f,b^2=f,e^2=h,f^2=h,b^a=bd,c^a=ce,e^a=eh,g^a=gh,c^b=ch,e^b=egh,d^c=dg,e^c=eh,e^d=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | #152461, #152476, #152478, … |
| maximal normal subgroup of | #152461, #152476, #152478, … |
| maximal quotient of | #135940, #135941, #135958, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5957);