Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33517 |
| automorphism | Aut(G) order 16384 |
| 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 | C24 | D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
290 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 290 conjugacy classes of subgroups — too many to draw. Only its 113 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)(3,6)(4,5)(7,8)(9,10,15,20)(11,21,24,38)(12,19,14,17)(13,26,29,34)(16,31,33,23)(18,25,36,28)(22,39,30,35)(27,37,40,32);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,22)(13,27)(14,30)(15,24)(17,32)(18,35)(19,37)(20,33)(21,26)(23,28)(25,31)(29,40)(34,38)(36,39);(9,12,15,14)(10,17,20,19)(11,22,24,30)(13,25,29,28)(16,32,33,37)(18,34,36,26)(21,35,38,39)(23,27,31,40)Polycyclic presentation:
2^8|a^2=h,c^2=e,d^2=h,f^2=gh,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fg,g^a=gh,f^c=fg,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #137438, #137442, #138980, … |
| maximal normal subgroup of | #137438, #137442, #138980, … |
| maximal quotient of | #493902, #494240, #494446, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26637);