Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
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 | |
745 conjugacy classes of subgroups; only the 195 normal subgroups are shown.
This group has 745 conjugacy classes of subgroups — too many to draw. Only its 195 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:
(2,4)(5,6,7,8)(9,10,13,11)(12,15,16,14)(18,20)(19,23);(10,11)(14,15)(17,18,21,20)(19,22,23,24);(5,6)(7,8)(9,11,13,10)(12,14,16,15)(17,19)(18,22)(20,24)(21,23);(1,2)(3,4)(5,6)(7,8)(9,12,13,16)(10,14,11,15)(17,20,21,18)(19,22,23,24)Polycyclic presentation:
2^8|a^2=ef,b^2=eg,c^2=g,d^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de52 conjugacy classes — view the full character table.
| maximal subgroup of | #104191, #104207, #104223, … |
| maximal normal subgroup of | #104191, #104207, #104223, … |
| maximal quotient of | #155667, #155683, #155841, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 9596);