Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24985 |
| automorphism | Aut(G) = #72261435 (order 2048) · Inn(G) = #555 · Out(G) = #120 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #555 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | #2572 |
| prime radical | #575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
276 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 276 conjugacy classes of subgroups — too many to draw. Only its 93 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,5,9,6,10,4,7)(3,11,12,16,13,8,14,15);(2,7,10,9)(3,12,13,14)(8,11)(15,16);(1,3)(2,8)(4,12)(5,14)(6,13)(7,15)(9,16)(10,11);(2,7,10,9)(8,15,11,16)Polycyclic presentation:
2^8|a^2=g,b^2=eg,d^2=f,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #151364, #151365, C2×#29232, … |
| maximal normal subgroup of | #151364, #151365, C2×#29232, … |
| maximal quotient of | #362387, #365995, #366269, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22220);