Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29730 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
431 conjugacy classes of subgroups; only the 159 normal subgroups are shown.
This group has 431 conjugacy classes of subgroups — too many to draw. Only its 159 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,8,11,9,7,10,12)(13,14,17,20)(15,21,23,18)(16,19,24,26)(22,27,28,25);(1,2,3,4)(6,7)(11,12)(13,15,17,23)(14,18,20,21)(16,22,24,28)(19,25,26,27);(2,4)(5,7,9,6)(8,11,10,12)(16,24)(19,26)(22,28)(25,27);(5,8,9,10)(6,11,7,12)(13,16,17,24)(14,19,20,26)(15,22,23,28)(18,25,21,27)Polycyclic presentation:
2^8|a^2=ef,b^2=eg,c^2=h,d^2=e,f^2=h,b^a=be,c^a=cf,c^b=cg,d^c=de,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #103959, #103963, #103967, … |
| maximal normal subgroup of | #103959, #103963, #103967, … |
| maximal quotient of | #331516, #331522, #331523, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23267);