Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
611 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 611 conjugacy classes of subgroups — too many to draw. Only its 183 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) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(5,6,7,8)(10,13)(14,16)(17,18,21,20)(19,23,24,22);(1,2,4,3)(5,6)(7,8)(10,13)(14,16);(1,2,4,3)(9,10)(11,14)(12,13)(15,16)(17,19,21,24)(18,22,20,23);(1,3,4,2)(9,11,12,15)(10,14,13,16)(17,20,21,18)(19,23,24,22)Polycyclic presentation:
2^8|a^2=efg,b^2=h,c^2=fg,d^2=f,b^a=be,c^a=cf,d^a=dh,c^b=cg64 conjugacy classes — view the full character table.
| maximal subgroup of | #101120, #101136, #103314, … |
| maximal normal subgroup of | #101120, #101136, #103314, … |
| maximal quotient of | #154159, #154286, #154309, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 8078);