Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#1967 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
823 conjugacy classes of subgroups; only the 247 normal subgroups are shown.
This group has 823 conjugacy classes of subgroups — too many to draw. Only its 247 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(4,6)(7,8,11,14,10,13,17,19)(9,12,16,21,15,20,18,22);(7,9,11,18,10,15,17,16)(8,12,19,21,13,20,14,22);(3,4,5,6)(8,13)(12,20)(14,19)(21,22)64 conjugacy classes — view the full character table.
| maximal subgroup of | #107164, #107248, #110093, … |
| maximal normal subgroup of | #107164, #107248, #110093, … |
| maximal quotient of | C2×#9651, #204167, C2×#9652, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14205);