Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) order 393216 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C4×#550 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator | C22 | C24×C4 |
| Frattini, socle, cosocle | C23 | C25 |
| prime radical | C2×#550 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1710 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2,3,4);(5,6)(7,8)(10,14)(15,16);(9,10,13,14)(11,15,12,16);(6,8)(9,11,13,12)(10,15,14,16);(9,12,13,11)(10,15,14,16)100 conjugacy classes — view the full character table.
| Fitting subgroup of | C4×C4⋊(#41⋊C3) |
| maximal subgroup of | #97659, C4×#1621, C4×#1631, … |
| maximal normal subgroup of | #97659, C4×#1621, C4×#1631, … |
| maximal quotient of | C4×#2354, C4×#2341, C4×Q8×C4⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27148);