Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#2053 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #124 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | #394 |
| prime radical | C2×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
683 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 683 conjugacy classes of subgroups — too many to draw. Only its 167 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4)(5,14)(6,19)(7,24)(8,12)(9,13)(10,16)(11,21)(15,25)(17,27)(18,26)(20,28)(22,29)(23,32)(30,33)(31,34);(3,5)(4,10)(6,20)(7,17)(8,16)(9,18)(11,28)(12,26)(13,27)(14,24)(15,22)(19,25)(21,31)(23,30)(29,33)(32,34);(3,6,7,21,9,23,8,22)(4,11,12,19,13,29,24,32)(5,15,16,30,18,31,17,20)(10,25,14,28,27,34,26,33)52 conjugacy classes — view the full character table.
| maximal subgroup of | #137196, #137197, #137198, … |
| maximal normal subgroup of | #137196, #137197, #137198, … |
| maximal quotient of | C2×#10512, C2×#10514, C2×#10516, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16522);