Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33522 |
| automorphism | Aut(G) order 16384 |
| 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 | #3155 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ #405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
356 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 356 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) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(7,15)(8,18)(9,20)(11,23)(13,28)(14,27)(16,33)(17,32)(19,22)(21,31)(24,35)(25,36)(26,34)(29,30);(1,2,3,4)(5,6)(7,11)(8,13)(9,14)(10,12)(15,23)(16,25)(17,24)(18,28)(19,29)(20,27)(21,26)(22,30)(31,34)(32,35)(33,36);(5,7,10,17)(6,11,12,24)(8,19,21,33)(9,22,20,16)(13,29,26,36)(14,30,27,25)(15,31,32,18)(23,34,35,28);(6,12)(11,24)(13,26)(14,27)(23,35)(25,30)(28,34)(29,36)Polycyclic presentation:
2^8|b^2=e,c^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 | #500750, #500756, #501022, … |
| maximal normal subgroup of | #500750, #500756, #501022, … |
| maximal quotient of | #490436, #490442, #490460, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26575);