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) = C27 |
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 | |
1119 conjugacy classes of subgroups; only the 207 normal subgroups are shown.
This group has 1119 conjugacy classes of subgroups — too many to draw. Only its 207 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) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(6,7)(9,10,11,12)(13,14,16,18)(15,17,19,20);(5,6,8,7)(9,10)(11,12)(15,19)(17,20);(9,10)(11,12)(13,15)(14,17)(16,19)(18,20);(1,2)(3,4)(5,7,8,6)(9,10)(11,12)(13,14)(15,17)(16,18)(19,20)Polycyclic presentation:
2^8|a^2=fg,b^2=ef,d^2=ef,b^a=be,c^a=cf,d^a=dh,c^b=cg64 conjugacy classes — view the full character table.
| maximal subgroup of | #101119, #101135, #103302, … |
| maximal normal subgroup of | #101119, #101135, #103302, … |
| maximal quotient of | #154117, #154204, #154286, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 7856);