Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| socle | C22 | C22×D8 |
| prime radical | #3147 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
295 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 295 conjugacy classes of subgroups — too many to draw. Only its 119 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) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,24)(12,17)(13,19)(14,33)(15,35)(16,23)(18,27)(20,31)(21,34)(22,32)(25,37)(26,38)(28,36)(29,40)(30,39);(2,5)(6,8)(9,11,15,26,13,25,14,12,16,27,31,28,32,29,34,30)(10,17,21,37,22,38,20,24,23,39,33,40,19,36,35,18);(1,3,4,7)(2,6,5,8)(9,12,14,29,32,26,15,27,16,30,34,25,13,28,31,11)(10,18,20,38,19,40,21,17,23,24,35,36,22,37,33,39);(9,13,16,32)(10,19,23,22)(11,25,27,29)(12,28,30,26)(14,31,34,15)(17,36,39,38)(18,40,24,37)(20,21,35,33)Polycyclic presentation:
2^8|b^2=f,c^2=e,d^2=h,e^2=gh,f^2=g,g^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fgh,g^a=gh,c^b=ch52 conjugacy classes — view the full character table.
| maximal subgroup of | #137512, #137572, #141266, … |
| maximal normal subgroup of | #137512, #137572, #141266, … |
| maximal quotient of | #494085, #494907, #495120, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26689);