Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
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 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
633 conjugacy classes of subgroups; only the 179 normal subgroups are shown.
This group has 633 conjugacy classes of subgroups — too many to draw. Only its 179 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:
(5,6,8,10)(7,11,12,9)(13,14)(15,22)(16,20)(17,26)(18,21)(19,24)(23,27)(25,28);(1,2)(3,4)(5,7,8,12)(6,9,10,11);(2,4)(13,15,18,25)(14,19,21,27)(16,24,17,23)(20,28,26,22);(13,16,18,17)(14,20,21,26)(15,23,25,24)(19,22,27,28)Polycyclic presentation:
2^8|a^2=e,b^2=e,c^2=h,d^2=h,f^2=h,b^a=be,c^a=cf,f^a=fh,c^b=cg,d^c=dh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #101284, #101287, #103433, … |
| maximal normal subgroup of | #101284, #101287, #103433, … |
| maximal quotient of | #237746, #237750, #237756, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16443);