Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60753 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | C2×#540 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
877 conjugacy classes of subgroups; only the 439 normal subgroups are shown.
This group has 877 conjugacy classes of subgroups — too many to draw. Only its 439 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(19,27)(20,21)(23,30)(25,26)(28,29)(31,32);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,18)(19,23)(20,25)(21,26)(22,24)(27,30)(28,31)(29,32);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,19)(18,23)(20,27)(21,29)(22,28)(24,31)(25,30)(26,32);(2,8)(6,9)(7,16)(14,15)(18,24)(23,31)(25,26)(30,32);(17,20,22,21)(18,25,24,26)(19,27,28,29)(23,30,31,32)Polycyclic presentation:
2^8|b^2=f,c^2=f,e^2=h,g^2=h,b^a=bf,c^a=cg,d^a=df,e^a=eh,g^a=gh,c^b=cf,d^b=dh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #133275, #246384, #246386, … |
| maximal normal subgroup of | #133275, #246384, #246386, … |
| maximal quotient of | #6374368, #6374455, #6374988, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53856);