Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | Aut(G) order 65536 |
| 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 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
403 conjugacy classes of subgroups; only the 147 normal subgroups are shown.
This group has 403 conjugacy classes of subgroups — too many to draw. Only its 147 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:
(1,2,3,4)(5,6,9,12)(7,11,8,10)(13,14,18,21,19,22,17,15)(16,23,25,24,26,28,27,20);(5,7,9,8)(6,10,12,11)(13,15,17,22,19,21,18,14)(16,24,27,23,26,20,25,28);(2,4)(13,16,19,26)(14,20,22,24)(15,23,21,28)(17,25,18,27);(5,8,9,7)(6,11,12,10)(13,17,19,18)(14,15,22,21)(16,25,26,27)(20,23,24,28)Polycyclic presentation:
2^8|a^2=ef,b^2=egh,c^2=h,d^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #501543, #501544, #501547, … |
| maximal normal subgroup of | #501543, #501544, #501547, … |
| maximal quotient of | #323269, #323277, #323286, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21523);