Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C22×#127 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1655 conjugacy classes of subgroups; only the 495 normal subgroups are shown.
This group has 1655 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) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6,9,7)(8,10,12,11);(1,2)(3,4)(5,6,9,7)(8,11,12,10)(13,14)(15,16)(17,18)(19,20);(6,7)(10,11)(14,18)(15,19);(5,7,9,6)(8,11,12,10)(13,15,17,19)(14,16,18,20);(5,8)(6,10)(7,11)(9,12)(13,16)(14,19)(15,18)(17,20)Polycyclic presentation:
2^8|a^2=g,b^2=g,d^2=h,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 | #97466, #129426, #133309, … |
| maximal normal subgroup of | #97466, #129426, #133309, … |
| maximal quotient of | #515303, #515363, #515522, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28542);