Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33522 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | C2×#508 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ #405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
347 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 347 conjugacy classes of subgroups — too many to draw. Only its 119 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,11,16)(7,12,19,30)(8,14,10,26)(9,22,25,36)(13,31,33,18)(15,21,34,24)(17,27,20,29)(23,28,35,32);(1,2,3,4)(5,7)(6,12)(8,20)(9,18)(10,17)(11,19)(13,22)(14,29)(15,28)(16,30)(21,35)(23,24)(25,31)(26,27)(32,34)(33,36);(5,7)(6,13)(8,17)(9,23)(10,20)(11,19)(12,22)(14,28)(15,29)(16,33)(18,24)(21,31)(25,35)(26,32)(27,34)(30,36);(5,8,11,10)(6,14,16,26)(7,17,19,20)(9,21,25,24)(12,27,30,29)(13,28,33,32)(15,22,34,36)(18,23,31,35)Polycyclic presentation:
2^8|a^2=h,b^2=e,d^2=h,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 | #501033, #501035, #501036, … |
| maximal normal subgroup of | #501033, #501035, #501036, … |
| maximal quotient of | #490439, #490445, #490463, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26578);