Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2883 |
| automorphism | Aut(G) order 393216 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
| direct product | C2×#2889 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C22 | C24×C4 |
| Frattini, cosocle | C2×C4 | C25 |
| prime radical | C22×2−1+4 | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
2385 conjugacy classes of subgroups; only the 991 normal subgroups are shown.
This group has 2385 conjugacy classes of subgroups — too many to draw. Only its 991 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4,7,12,8,11,20,25)(5,13,15,26,16,9,19,22)(6,10,18,24,17,23,30,33)(14,27,29,34,28,21,31,32);(3,5,8,16)(4,9,11,13)(6,14,17,28)(7,19,20,15)(10,21,23,27)(12,26,25,22)(18,31,30,29)(24,34,33,32);(3,6)(4,10)(5,14)(7,18)(8,17)(9,21)(11,23)(12,24)(13,27)(15,29)(16,28)(19,31)(20,30)(22,32)(25,33)(26,34);(6,17)(10,23)(14,28)(18,30)(21,27)(24,33)(29,31)(32,34)82 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×#2889⋊C3 |
| maximal subgroup of | #146579, #146583, #146586, … |
| maximal normal subgroup of | #146579, #146583, #146586, … |
| maximal quotient of | C2×#17351, C2×#17352, #6342547, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53063);