Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2883 |
| automorphism | Aut(G) order 98304 |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C22 | C23×C8 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
581 conjugacy classes of subgroups; only the 263 normal subgroups are shown.
This group has 581 conjugacy classes of subgroups — too many to draw. Only its 263 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,3,5,4,6,7,8)(9,10)(11,19,22,15)(12,16)(13,17)(14,18)(20,33,36,27)(21,34,38,28)(23,29)(24,30)(25,31)(26,32)(35,40,37,39);(9,11)(10,15)(12,20)(13,21)(14,22)(16,27)(17,28)(18,19)(23,35)(24,36)(25,37)(26,38)(29,39)(30,33)(31,40)(32,34);(9,12,14,24)(10,16,18,30)(11,20,22,36)(13,25,26,23)(15,27,19,33)(17,31,32,29)(21,37,38,35)(28,40,34,39);(9,13,14,26)(10,17,18,32)(11,21,22,38)(12,23,24,25)(15,28,19,34)(16,29,30,31)(20,35,36,37)(27,39,33,40)Polycyclic presentation:
2^8|a^2=f,c^2=g,d^2=g,f^2=h,b^a=be,e^a=eg,f^b=fg,d^c=dg82 conjugacy classes — view the full character table.
| Fitting subgroup of | #17352⋊C3 |
| maximal subgroup of | #137689, #137691, #137731, … |
| maximal normal subgroup of | #137689, #137691, #137731, … |
| maximal quotient of | #160506, #160510, #160516, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10340);