Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2846 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1251 conjugacy classes of subgroups; only the 631 normal subgroups are shown.
This group has 1251 conjugacy classes of subgroups — too many to draw. Only its 631 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) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,16,18)(15,19,20,17);(6,9)(10,12);(13,15,16,20)(14,17,18,19);(1,2,3,4);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^8|a^2=g,c^2=g,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=dh88 conjugacy classes — view the full character table.
| maximal subgroup of | #111238, #112604, #122009, … |
| maximal normal subgroup of | #111238, #112604, #122009, … |
| maximal quotient of | #607577, #607618, #607619, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29633);