Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C22×#120 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
2025 conjugacy classes of subgroups; only the 495 normal subgroups are shown.
This group has 2025 conjugacy classes of subgroups — too many to draw. Only its 495 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,3)(13,14)(15,16)(17,18)(19,20);(1,2,4,3)(5,6)(7,11)(8,9)(10,12)(13,15,17,19)(14,16,18,20);(1,3,4,2)(6,9)(10,11)(14,18)(15,19);(5,6)(7,10)(8,9)(11,12)(13,14)(15,16)(17,18)(19,20);(5,7)(6,10)(8,12)(9,11)(13,16)(14,15)(17,20)(18,19)Polycyclic presentation:
2^8|b^2=g,c^2=f,b^a=bf,c^a=cg,c^b=ch,e^b=eh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #97553, #129435, #131986, … |
| maximal normal subgroup of | #97553, #129435, #131986, … |
| maximal quotient of | #515290, #515326, #515561, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28355);