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) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #102 |
| commutator | C2×C8 | C42 |
| Frattini, cosocle | C23×C8 | C22 |
| prime radical | C24 | D8 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
201 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 201 conjugacy classes of subgroups — too many to draw. Only its 67 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) = 64 |
A minimal faithful permutation representation — 2 generators on 28 points:
(1,2,3,4)(5,6)(7,9)(8,11)(10,12)(13,14)(15,18)(16,25)(17,21)(19,22)(20,23)(24,27)(26,28);(6,9,11,10)(14,18)(15,22)(16,23)(19,20)(21,27)(24,28)(25,26)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fh,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×#7258, #127510, #127511, … |
| maximal normal subgroup of | C2×#7258, #127510, #127511, … |
| maximal quotient of | #93031, #93034, #93043, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 246);