Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×#2242 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
305 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 305 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) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(3,4)(5,8)(6,7)(9,10)(11,12)(13,25)(14,19)(15,33)(16,36)(17,37)(18,24)(20,29)(21,26)(22,35)(23,34)(27,39)(28,38)(30,42)(31,41)(32,40);(4,7)(8,10)(11,13,26,31,17,28,35,32,18,29,16,27,34,30,15,14)(12,19,33,42,23,39,36,20,24,40,22,38,37,41,21,25);(3,5,6,9)(4,8,7,10)(11,14,15,30,34,27,16,29,18,32,35,28,17,31,26,13)(12,20,21,39,37,42,22,19,24,25,36,41,23,38,33,40)76 conjugacy classes — view the full character table.
| maximal subgroup of | #141531, #141532, C4×#2242, … |
| maximal normal subgroup of | #141531, #141532, C4×#2242, … |
| maximal quotient of | C2×#12889, C2×#13147, C2×#13148, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26684);