Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#1957 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×#127 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1207 conjugacy classes of subgroups; only the 367 normal subgroups are shown.
This group has 1207 conjugacy classes of subgroups — too many to draw. Only its 367 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(4,6)(7,8,9,10)(12,15)(13,16)(17,18);(8,10)(11,12,16,18,14,17,13,15);(3,4,5,6)88 conjugacy classes — view the full character table.
| Frattini subgroup of | #72279955, #72279985 |
| maximal subgroup of | #108334, #110171, #110263, … |
| maximal normal subgroup of | #108334, #110171, #110263, … |
| maximal quotient of | C2×#9638, C2×#9644, #204059, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14181);