Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #102 |
| commutator | C2×C8 | C42 |
| Frattini, cosocle | C23×C8 | C22 |
| prime radical | C24 | SD16 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
171 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 171 conjugacy classes of subgroups — too many to draw. Only its 59 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 — 2 generators on 28 points:
(1,2,3,4)(5,6)(7,9)(8,11)(10,12)(14,19)(15,23)(16,17)(20,27)(21,24)(22,28)(25,26);(6,9,11,10)(13,14,16,20,18,22,17,21)(15,24,23,28,26,27,25,19)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fh,e^2=h,f^2=h,g^2=h,b^a=bc,c^a=cf,e^a=efgh,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7259, #127514, #127515, … |
| maximal normal subgroup of | C2×#7259, #127514, #127515, … |
| maximal quotient of | #93032, #93033, #93044, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 247);