Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| 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 | #3283 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
274 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 274 conjugacy classes of subgroups — too many to draw. Only its 101 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:
(1,2,4,6)(3,7,8,5)(9,10,15,20)(11,21,22,38)(12,19,14,17)(13,26,29,34)(16,31,32,24)(18,25,36,28)(23,39,30,35)(27,37,40,33);(1,3)(2,5)(4,8)(6,7)(11,22)(16,32)(21,38)(23,30)(24,31)(27,40)(33,37)(35,39);(9,11)(10,16)(12,23)(13,27)(14,30)(15,22)(17,33)(18,35)(19,37)(20,32)(21,26)(24,28)(25,31)(29,40)(34,38)(36,39);(3,8)(5,7)(9,12,15,14)(10,17,20,19)(11,23,22,30)(13,25,29,28)(16,33,32,37)(18,34,36,26)(21,35,38,39)(24,27,31,40)Polycyclic presentation:
2^8|a^2=eh,d^2=h,f^2=gh,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fg,g^a=gh,c^b=ch,d^b=de,f^c=fg,g^c=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | #491749, #491756, #491765, … |
| maximal normal subgroup of | #491749, #491756, #491765, … |
| maximal quotient of | #497387, #497547, #497725, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26886);