Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10320 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator, Frattini, cosocle | C23×C4 | C23 |
| prime radical | C25 | D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
455 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 455 conjugacy classes of subgroups — too many to draw. Only its 61 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) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10,13,16)(11,17,19,24)(12,20,22,15)(14,21,23,18);(1,3)(2,5)(4,7)(6,8)(9,11,13,19)(10,14,16,23)(12,21,22,18)(15,24,20,17);(5,8)Polycyclic presentation:
2^8|a^2=g,b^2=g,d^2=g,b^a=bd,c^a=ce,d^a=dg,f^a=fh,c^b=cf,d^b=dg,e^b=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | #93794, #93834, #122981, … |
| maximal normal subgroup of | #93794, #93834, #122981, … |
| maximal quotient of | #102457, #102458, #102459, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3312);