Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#2099 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
499 conjugacy classes of subgroups; only the 215 normal subgroups are shown.
This group has 499 conjugacy classes of subgroups — too many to draw. Only its 215 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(3,4,7,9)(5,10,6,8)(11,12)(13,15)(14,16)(17,18);(3,5,7,6)(4,8,9,10)(11,13,14,17)(12,15,16,18)(19,20)(21,23)(22,25)(24,26);(3,6,7,5)(4,8,9,10)(12,16)(15,18)(20,23)(21,26)(24,25)64 conjugacy classes — view the full character table.
| maximal subgroup of | #99646, #99647, #103382, … |
| maximal normal subgroup of | #99646, #99647, #103382, … |
| maximal quotient of | C2×#10687, #244081, #244098, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16729);