Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C25 |
| commutator | C22 | C24×C4 |
| prime radical | C22×#127 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1695 conjugacy classes of subgroups; only the 719 normal subgroups are shown.
This group has 1695 conjugacy classes of subgroups — too many to draw. Only its 719 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:
(1,2,3,4)(5,6,9,7)(8,11,12,10)(13,14)(15,17)(16,19)(18,20);(6,7)(10,11)(14,15)(18,19);(13,14,17,15)(16,18,20,19);(5,7,9,6)(8,11,12,10)(13,15,17,14)(16,19,20,18);(5,8)(6,10)(7,11)(9,12)(13,16)(14,18)(15,19)(17,20)Polycyclic presentation:
2^8|a^2=h,c^2=g,d^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df76 conjugacy classes — view the full character table.
| maximal subgroup of | #107220, #107301, #107339, … |
| maximal normal subgroup of | #107220, #107301, #107339, … |
| maximal quotient of | #513784, #513807, #513833, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27352);