Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33607 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | #3176 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ #405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
254 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 254 conjugacy classes of subgroups — too many to draw. Only its 103 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)(11,19)(12,23)(13,24)(15,26)(17,32)(18,31)(20,25)(21,36)(22,35)(27,30)(28,37)(29,40)(33,34)(38,39);(2,5)(6,8)(9,10,14,16)(11,15,22,28)(12,17,25,30)(13,18,24,31)(19,26,35,37)(20,27,23,32)(21,29,33,38)(34,39,36,40);(9,11,20,36,13,21,12,19,14,22,23,34,24,33,25,35)(10,15,27,40,18,29,17,26,16,28,32,39,31,38,30,37);(1,3,4,7)(2,6,5,8)(10,16)(15,28)(17,30)(18,31)(26,37)(27,32)(29,38)(39,40)Polycyclic presentation:
2^8|b^2=h,c^2=fgh,d^2=e,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,d^b=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #141373, #141376, #491119, … |
| maximal normal subgroup of | #141373, #141376, #491119, … |
| maximal quotient of | #494207, #495303, #495310, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26731);