Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17405 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#540 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
541 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 541 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) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(9,10,13,16,12,15,20,23)(11,17,19,24,18,14,21,22);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,18)(13,19)(15,17)(16,22)(20,21)(23,24);(2,5)(6,8)(10,15)(11,18)(13,20)(22,24);(1,3,4,7)(2,6,5,8)(10,15)(11,18)(16,23)(19,21)Polycyclic presentation:
2^8|a^2=g,d^2=eh,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=dh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #214912, #214914, #214916, … |
| maximal normal subgroup of | #214912, #214914, #214916, … |
| maximal quotient of | #216194, #216196, #216200, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14870);