Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2069 |
| automorphism | Aut(G) order 49152 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#2070 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #396 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
475 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 475 conjugacy classes of subgroups — too many to draw. Only its 167 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 — 4 generators on 34 points:
(1,2);(3,4,9,12,8,13,7,21)(5,14,23,25,17,30,16,10)(6,11,22,26,20,27,19,28)(15,29,32,24,18,33,31,34);(3,5,7,16,8,17,9,23)(4,10,12,14,13,25,21,30)(6,18,19,32,20,15,22,31)(11,24,26,33,27,34,28,29);(3,6,8,20)(4,11,13,27)(5,15,17,18)(7,22,9,19)(10,24,25,34)(12,28,21,26)(14,29,30,33)(16,32,23,31)46 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×#2070⋊C3 |
| maximal subgroup of | C4×#2070, #377687, #377935, … |
| maximal normal subgroup of | C4×#2070, #377687, #377935, … |
| maximal quotient of | C2×#10604, C2×#10608, C2×#10606, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16609);