Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33764 |
| 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 |
| prime radical | C23 | C2×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ #405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
221 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 221 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,22,27)(19,23,28,36)(20,24,33,40)(21,34,32,26)(25,41,39,31)(29,35,42,46)(30,45,43,38)(37,48,47,44);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19)(18,23)(20,29)(21,30)(22,28)(24,35)(25,37)(26,38)(27,36)(31,44)(32,43)(33,42)(34,45)(39,47)(40,46)(41,48);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(18,24)(20,31)(21,32)(23,35)(25,26)(27,40)(29,44)(30,43)(33,41)(34,39)(36,46)(37,38)(42,48)(45,47);(2,8)(6,9)(7,16)(14,15)(19,28)(23,36)(29,42)(30,43)(35,46)(37,47)(38,45)(44,48)Polycyclic presentation:
2^8|a^2=h,b^2=e,c^2=e,f^2=gh,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fg,g^a=gh,c^b=ce,d^b=dh,f^c=fg,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#33808, #7623414, #7623422, … |
| maximal normal subgroup of | C2×#33808, #7623414, #7623422, … |
| maximal quotient of | #496447, #496490, #496744, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26796);