Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C8 | C24 |
| commutator, socle | C22 | C23×C8 |
| prime radical | C24 | C2×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
353 conjugacy classes of subgroups; only the 263 normal subgroups are shown.
This group has 353 conjugacy classes of subgroups — too many to draw. Only its 263 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)(5,6,8,13,9,14,18,26,10,12,19,24,20,25,30,34)(7,11,16,22,17,23,29,33,15,21,27,31,28,32,35,36);(1,2,3,4)(5,7)(6,11)(8,16)(9,17)(10,15)(12,21)(13,22)(14,23)(18,29)(19,27)(20,28)(24,31)(25,32)(26,33)(30,35)(34,36);(7,15)(11,21)(16,27)(17,28)(22,31)(23,32)(29,35)(33,36);(6,12)(11,21)(13,24)(14,25)(22,31)(23,32)(26,34)(33,36)Polycyclic presentation:
2^8|a^2=f,b^2=e,f^2=g,g^2=h,b^a=be,d^a=dh,c^b=ch88 conjugacy classes — view the full character table.
| maximal subgroup of | #127406, #127408, #135812, … |
| maximal normal subgroup of | #127406, #127408, #135812, … |
| maximal quotient of | #483318, #483331, #483347, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26385);