Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) = C22×D4×#525 (order 2048) · Inn(G) = C2×D4 · Out(G) = C23×#41 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×D4 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C2×C22⋊C8 |
| prime radical | C2×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
225 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 225 conjugacy classes of subgroups — too many to draw. Only its 97 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,9,14,10,15,24,32,11,16,25,33,8,13,22,30)(7,17,19,34,18,28,35,31,20,29,23,36,21,12,26,27);(1,2,3,4);(5,7)(6,12)(8,21)(9,23)(10,18)(11,20)(13,29)(14,31)(15,17)(16,28)(19,25)(22,35)(24,26)(27,33)(30,34)(32,36)Polycyclic presentation:
2^8|a^2=f,b^2=d,e^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,e^c=eh,f^c=fh76 conjugacy classes — view the full character table.
| maximal subgroup of | #129514, #129515, #129535, … |
| maximal normal subgroup of | #129514, #129515, #129535, … |
| maximal quotient of | #134220, #134221, #134248, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5775);