Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #15257 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
847 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 847 conjugacy classes of subgroups — too many to draw. Only its 167 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) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6,8,7)(9,10)(11,14)(12,13)(15,16)(18,19)(20,24);(6,7)(10,12)(14,15)(17,18,21,19)(20,23,24,22);(5,6,8,7)(9,11,13,16)(10,14,12,15)(17,19,21,18)(20,23,24,22);(1,2)(3,4)(5,7,8,6)(9,12,13,10)(11,15,16,14)(17,20)(18,22)(19,23)(21,24)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=e,d^2=ef,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de58 conjugacy classes — view the full character table.
| maximal subgroup of | #125847, #126168, #126223, … |
| maximal normal subgroup of | #125847, #126168, #126223, … |
| maximal quotient of | #154714, #154879, #154979, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 8557);