Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2883 |
| automorphism | Aut(G) order 786432 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
| direct product | C2×#2884 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C22 | C24×C4 |
| Frattini, cosocle | C23 | C25 |
| prime radical | C22×2−1+4 | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C23 ▷ C2 ▷ C1 | |
2409 conjugacy classes of subgroups; only the 991 normal subgroups are shown.
This group has 2409 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)(5,13,16,9)(6,10)(7,11)(8,12)(14,27,30,21)(15,28,32,22)(17,23)(18,24)(19,25)(20,26)(29,34,31,33);(3,5)(4,9)(6,14)(7,15)(8,16)(10,21)(11,22)(12,13)(17,29)(18,30)(19,31)(20,32)(23,33)(24,27)(25,34)(26,28);(3,6,8,18)(4,10,12,24)(5,14,16,30)(7,19,20,17)(9,21,13,27)(11,25,26,23)(15,31,32,29)(22,34,28,33);(3,7,8,20)(4,11,12,26)(5,15,16,32)(6,17,18,19)(9,22,13,28)(10,23,24,25)(14,29,30,31)(21,33,27,34)82 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×#2884⋊C3 |
| maximal subgroup of | #146576, #146585, #146588, … |
| maximal normal subgroup of | #146576, #146585, #146588, … |
| maximal quotient of | C2×#17352, C2×#17353, #6342545, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53057);