Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C25 |
| commutator | C22 | C24×C4 |
| prime radical | C23×#41 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
2715 conjugacy classes of subgroups; only the 999 normal subgroups are shown.
This group has 2715 conjugacy classes of subgroups — too many to draw. Only its 999 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6,7,8)(9,10)(11,17)(12,20)(13,14)(15,19)(16,21)(18,22)(23,24);(10,14)(15,17)(16,20)(22,23);(1,2)(3,4)(5,6)(7,8)(9,11,13,19)(10,15,14,17)(12,18,21,24)(16,22,20,23);(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24);(9,12)(10,16)(11,18)(13,21)(14,20)(15,22)(17,23)(19,24)Polycyclic presentation:
2^8|a^2=h,c^2=f,b^a=bf,c^a=cg,e^a=ef,d^b=df88 conjugacy classes — view the full character table.
| maximal subgroup of | #105838, #106738, #106772, … |
| maximal normal subgroup of | #105838, #106738, #106772, … |
| maximal quotient of | #513546, #513548, #513677, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27160);