Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32932 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
397 conjugacy classes of subgroups; only the 155 normal subgroups are shown.
This group has 397 conjugacy classes of subgroups — too many to draw. Only its 155 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 — 4 generators on 28 points:
(1,2)(3,4)(5,6,8,10)(7,11,12,9)(15,21)(16,22)(18,24)(20,25)(23,27)(26,28);(5,7,8,12)(6,9,10,11)(13,14,17,19)(15,18,23,26)(16,20,22,25)(21,24,27,28);(13,15,22,27,17,23,16,21)(14,18,25,28,19,26,20,24);(2,4)(14,19)(18,26)(20,25)(24,28)Polycyclic presentation:
2^8|a^2=e,b^2=eh,c^2=fh,f^2=h,b^a=be,c^a=cf,d^a=dg,f^a=fh,d^b=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #125205, #125462, #138101, … |
| maximal normal subgroup of | #125205, #125462, #138101, … |
| maximal quotient of | #352458, #352459, #352466, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26204);