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) = C24×C4 |
| direct product | C2×#1998 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
563 conjugacy classes of subgroups; only the 247 normal subgroups are shown.
This group has 563 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) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(3,4)(5,9)(6,7)(8,10)(11,12,15,18,14,17,19,25)(13,16,21,24,20,23,22,26);(4,7)(8,9)(11,13,19,21,14,20,15,22)(12,16,18,26,17,23,25,24);(3,5,6,10)(4,8,7,9)(12,17)(16,23)(18,25)(24,26)64 conjugacy classes — view the full character table.
| maximal subgroup of | #107165, #107249, #110094, … |
| maximal normal subgroup of | #107165, #107249, #110094, … |
| maximal quotient of | C2×#9880, #206872, C2×#9881, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14324);