Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33522 |
| 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 | |
279 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 279 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,26)(14,19)(17,23)(18,24)(20,28)(21,27)(22,36)(25,32)(29,35)(30,34)(31,37)(33,38)(39,40);(2,5)(6,8)(9,11,14,20)(10,15,19,28)(12,21,25,33)(13,22,24,34)(16,27,32,38)(17,29,31,39)(18,30,26,36)(23,35,37,40);(10,16)(12,23)(13,24)(15,27)(17,18)(19,32)(21,35)(22,34)(25,37)(26,31)(28,38)(29,30)(33,40)(36,39);(1,3,4,7)(2,6,5,8)(11,20)(15,28)(21,33)(22,34)(27,38)(29,39)(30,36)(35,40)Polycyclic presentation:
2^8|b^2=h,d^2=e,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,d^b=dh,f^c=fg,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#33679, #7623095, #7623103, … |
| maximal normal subgroup of | C2×#33679, #7623095, #7623103, … |
| maximal quotient of | #494037, #494972, #494974, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26667);