Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60388 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#120 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
2437 conjugacy classes of subgroups; only the 557 normal subgroups are shown.
This group has 2437 conjugacy classes of subgroups — too many to draw. Only its 557 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 5 generators on 16 points:
(2,6)(4,11)(7,13)(8,12)(10,15)(14,16);(1,2)(3,7)(4,12)(5,8)(6,11)(9,14)(10,16)(13,15);(4,11)(6,12)(10,15)(13,16);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,15)(12,16);(3,9)(7,14)(10,15)(13,16)Polycyclic presentation:
2^8|f^2=h,b^a=bf,c^a=cg,f^a=fh,f^b=fh,g^b=gh,f^c=fh,e^d=eh58 conjugacy classes — view the full character table.
| Fitting subgroup of | #60388⋊C3 |
| maximal subgroup of | #152027, #499881, #499886, … |
| maximal normal subgroup of | #152027, #499881, #499886, … |
| maximal quotient of | #6350397, #6350398, #6350399, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53376);