Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13304 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×D8 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
181 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 181 conjugacy classes of subgroups — too many to draw. Only its 61 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) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(14,18)(19,26)(20,25)(24,27);(6,9)(10,12)(14,19)(16,23)(17,22)(18,26)(20,27)(21,28)(24,25);(2,4)(5,7,8,11)(6,10,9,12)(13,14)(15,18)(16,24)(17,25)(19,23)(20,22)(21,27)(26,28)Polycyclic presentation:
2^8|a^2=eh,c^2=d,f^2=gh,g^2=h,b^a=bd,c^a=ce,c^b=cf,f^b=fg,g^b=gh,f^c=fg,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13482, #499345, #499346, … |
| maximal normal subgroup of | C2×#13482, #499345, #499346, … |
| maximal quotient of | #146006, #146019, #146038, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6470);