Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C2×#2020 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
951 conjugacy classes of subgroups; only the 191 normal subgroups are shown.
This group has 951 conjugacy classes of subgroups — too many to draw. Only its 191 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)(7,8,13,18)(9,19,23,30)(10,24,28,33)(11,17,12,16)(14,22,31,21)(15,27,34,26)(20,35,37,29)(25,32,36,38);(4,6)(7,9)(8,14)(10,25)(11,22)(12,21)(13,23)(15,32)(16,30)(17,19)(18,31)(20,27)(24,35)(26,37)(28,36)(29,33)(34,38);(7,10,13,28)(8,15,18,34)(9,20,23,37)(11,27,12,26)(14,29,31,35)(16,33,17,24)(19,32,30,38)(21,25,22,36)52 conjugacy classes — view the full character table.
| maximal subgroup of | #137148, #137154, #142237, … |
| maximal normal subgroup of | #137148, #137154, #142237, … |
| maximal quotient of | C2×#9976, C2×#9979, #232444, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16425);