Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
| direct product | C2×#2113 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C23 |
| commutator | C22 | C23×C8 |
| Frattini, cosocle | C2×C8 | C24 |
| socle | C23 | C22×C8 |
| prime radical | C22×#41 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
595 conjugacy classes of subgroups; only the 271 normal subgroups are shown.
This group has 595 conjugacy classes of subgroups — too many to draw. Only its 271 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(3,4)(5,6)(7,8,10,14,11,15,20,26,12,16,21,27,22,28,32,36)(9,13,18,24,19,25,31,35,17,23,29,33,30,34,37,38);(4,6)(7,9)(8,13)(10,18)(11,19)(12,17)(14,24)(15,25)(16,23)(20,31)(21,29)(22,30)(26,35)(27,33)(28,34)(32,37)(36,38);(9,17)(13,23)(18,29)(19,30)(24,33)(25,34)(31,37)(35,38)112 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7207, C2×#7208, C2×#7228, … |
| maximal normal subgroup of | C2×#7207, C2×#7208, C2×#7228, … |
| maximal quotient of | C2×#11419, C2×#11441, C2×#11461, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26320);