Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23924 |
| automorphism | Aut(G) order 32768 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
433 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 433 conjugacy classes of subgroups — too many to draw. Only its 123 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)(7,11)(8,9)(10,12)(13,14)(15,17)(16,23)(18,22)(19,25)(20,26)(21,27)(24,28);(1,2,3,4)(6,9)(10,11)(13,15,17,14)(16,19,24,21)(18,22,26,20)(23,27,28,25);(2,4)(13,16,17,24)(14,19,15,21)(18,27,26,25)(20,28,22,23);(5,7,8,12)(6,10,9,11)(14,15)(19,21)(20,22)(23,28)Polycyclic presentation:
2^8|b^2=g,c^2=h,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #150479, #150482, #150819, … |
| maximal normal subgroup of | #150479, #150482, #150819, … |
| maximal quotient of | #316819, #316830, #316840, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21161);