Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33607 |
| automorphism | Aut(G) = C26×#442 (order 4096) · Inn(G) = C22×D8 · Out(G) = C24×C4 |
| 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 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ #405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
260 conjugacy classes of subgroups.
| 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)(12,21)(13,24)(15,26)(17,27)(18,30)(20,31)(22,38)(23,34)(25,37)(28,40)(29,35)(32,33)(36,39);(2,5)(6,8)(9,10)(11,20)(12,17)(13,18)(14,16)(15,23)(19,31)(21,27)(22,36)(24,30)(25,29)(26,34)(28,32)(33,40)(35,37)(38,39);(9,11,21,33,13,22,25,34,14,23,37,38,24,32,12,19)(10,15,27,39,18,28,29,31,16,20,35,40,30,36,17,26);(1,3,4,7)(2,6,5,8)(10,16)(15,20)(17,29)(18,30)(26,31)(27,35)(28,36)(39,40)Polycyclic presentation:
2^8|c^2=fg,d^2=e,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,c^b=ch,d^b=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#33744, #7623240, #7623249, … |
| maximal normal subgroup of | C2×#33744, #7623240, #7623249, … |
| maximal quotient of | #494208, #495304, #495306, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26732);