Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9946 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator, Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
276 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 276 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,6)(4,7)(5,8)(9,12)(10,13)(11,15)(14,16)(17,18,22,26)(19,27,30,31)(20,23,21,25)(24,28,32,29);(1,3,5,10)(2,6,8,13)(4,9,11,16)(7,14,15,12)(18,23)(20,21)(24,31)(25,26)(27,32)(28,29);(1,4)(2,7)(3,9)(5,11)(6,12)(8,15)(10,16)(13,14)(17,19,22,30)(18,24,26,32)(20,29,21,28)(23,31,25,27)Polycyclic presentation:
2^8|a^2=g,b^2=gh,c^2=g,d^2=g,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fgh,c^b=cf,d^b=dg,e^b=eg,d^c=dh,e^c=eg37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9961, #235044, #235046, … |
| maximal normal subgroup of | C2×#9961, #235044, #235046, … |
| maximal quotient of | #102800, #102813, #102824, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2949);