Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13304 |
| automorphism | Aut(G) = C27×#442 (order 8192) · Inn(G) = C2×D8 · Out(G) = C26×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×D8 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
175 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 175 conjugacy classes of subgroups — too many to draw. Only its 69 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(5,6,9,11)(7,12,8,10)(13,14)(15,19)(16,20)(17,22)(18,26)(21,24)(23,27)(25,28);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,17,18)(14,19,22,26)(16,23,25,24)(20,27,28,21);(2,4)(5,8,9,7)(6,10,11,12)(14,20)(15,18)(16,24)(19,21)(22,28)(23,25)(26,27)Polycyclic presentation:
2^8|a^2=d,b^2=dh,c^2=d,e^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=eg,g^a=gh,c^b=cf,e^c=eg,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13453, #499189, #499191, … |
| maximal normal subgroup of | C2×#13453, #499189, #499191, … |
| maximal quotient of | #145782, #145812, #145855, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6441);