Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) order 393216 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C24 |
| commutator | C22 | C22×C42 |
| prime radical | C25 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1077 conjugacy classes of subgroups; only the 379 normal subgroups are shown.
This group has 1077 conjugacy classes of subgroups — too many to draw. Only its 379 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6,8,7)(10,12)(13,14,17,16)(15,18,20,19);(5,6,8,7)(14,16)(18,19);(9,10)(11,12)(13,15,17,20)(14,18,16,19);(1,3,4,2)(5,7,8,6)(13,16,17,14)(15,19,20,18)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=e,d^2=g,b^a=be,c^a=cf,d^b=de100 conjugacy classes — view the full character table.
| Fitting subgroup of | #13809⋊C3 |
| maximal subgroup of | #219845, #219847, #219879, … |
| maximal normal subgroup of | #219845, #219847, #219879, … |
| maximal quotient of | C22⋊C4×#95, #153725, #153726, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6797);