Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) order 393216 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
959 conjugacy classes of subgroups; only the 467 normal subgroups are shown.
This group has 959 conjugacy classes of subgroups — too many to draw. Only its 467 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 — 5 generators on 24 points:
(1,2)(3,6)(4,7)(5,8)(10,13)(11,15)(17,18)(19,23)(20,22)(21,24);(1,3,6,2)(4,8,5,7)(9,10,12,13)(11,14,15,16)(17,19)(18,21)(20,24)(22,23);(1,4,6,5)(2,7,3,8)(9,11,12,15)(10,14,13,16)(18,22)(19,24);(2,3)(7,8)(10,13)(14,16)(19,24)(21,23);(1,5,6,4)(2,8,3,7)Polycyclic presentation:
2^8|b^2=fh,c^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df64 conjugacy classes — view the full character table.
| Fitting subgroup of | #52291⋊C3, #72227894, #72227895 |
| maximal subgroup of | #97525, #131465, #149484, … |
| maximal normal subgroup of | #97525, #131465, #149484, … |
| maximal quotient of | #520824, #520940, #521101, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45279);