Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23994 |
| automorphism | Aut(G) = C22×#511819 (order 2048) · Inn(G) = #535 · Out(G) = C25 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #535 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | #2598 |
| prime radical | #2010 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
354 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 354 conjugacy classes of subgroups — too many to draw. Only its 101 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)(3,10)(4,7)(5,9)(6,11)(8,13)(12,15)(14,16);(1,3,4,11,5,12,13,16)(2,6,8,10,9,14,7,15);(6,14)(10,15);(2,7,9,8)(6,15,14,10)Polycyclic presentation:
2^8|b^2=e,d^2=g,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gh,c^b=cg,f^b=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #139694, #139695, #139696, … |
| maximal normal subgroup of | #139694, #139695, #139696, … |
| maximal quotient of | #326603, #326604, #326605, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22372);