Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
519 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 519 conjugacy classes of subgroups — too many to draw. Only its 135 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:
(5,6,8,9)(7,10,11,12)(13,14,18,20,17,22,16,21)(15,23,26,19,24,28,25,27);(1,2)(3,4)(6,9)(10,12)(13,15)(14,19)(16,26)(17,24)(18,25)(20,23)(21,28)(22,27);(2,4)(14,20)(16,18)(19,23)(21,22)(25,26)(27,28);(5,7,8,11)(6,10,9,12)(15,24)(19,27)(23,28)(25,26)Polycyclic presentation:
2^8|a^2=eh,d^2=efh,e^2=h,f^2=h,b^a=be,c^a=cf,c^b=cg,d^b=dh,e^b=eh,f^b=fh,e^c=eh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #219493, #219513, #219526, … |
| maximal normal subgroup of | #219493, #219513, #219526, … |
| maximal quotient of | #244136, #244140, #244184, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16799);