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) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
747 conjugacy classes of subgroups; only the 155 normal subgroups are shown.
This group has 747 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) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(10,14)(12,15)(16,21)(17,22)(18,24)(20,23);(9,10,15,22,13,17,12,14)(11,18,23,21,19,16,20,24);(2,4)(9,11)(10,16)(12,20)(13,19)(14,21)(15,23)(17,18)(22,24);(6,8)(11,19)(16,18)(20,23)(21,24)Polycyclic presentation:
2^8|b^2=eh,e^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,c^b=ch,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #137968, #137971, #137984, … |
| maximal normal subgroup of | #137968, #137971, #137984, … |
| maximal quotient of | #347731, #347732, #347733, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25927);