Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#2194 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #447 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | C2×#576 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
565 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 565 conjugacy classes of subgroups — too many to draw. Only its 127 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);(3,4,9,15)(5,16,19,32)(6,20,24,31)(7,14,8,13)(10,26,29,17)(11,25,30,18)(12,22,27,23)(21,33,34,28);(3,5)(4,10)(6,21)(7,18)(8,25)(9,19)(11,27)(12,30)(13,28)(14,33)(15,29)(16,20)(17,23)(22,26)(24,34)(31,32);(3,5)(4,11)(6,17)(7,25)(8,18)(9,19)(10,27)(12,29)(13,32)(14,16)(15,30)(20,33)(21,23)(22,34)(24,26)(28,31)46 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#2194, #495716, #498167, … |
| maximal normal subgroup of | C4×#2194, #495716, #498167, … |
| maximal quotient of | C2×#12111, C2×#12113, #484218, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26523);