Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18132 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #392 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
480 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 480 conjugacy classes of subgroups — too many to draw. Only its 115 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) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7)(3,11)(4,14)(8,20)(9,18)(10,15)(13,25)(16,27)(19,22)(21,30)(23,28)(29,31);(1,2,5,10)(3,8,11,22)(4,15,14,7)(6,9,17,23)(12,20,24,19)(13,21,25,31)(16,28,27,18)(26,30,32,29);(4,16)(7,18)(12,26)(14,27)(15,28)(19,29)(20,30)(24,32);(1,3,6,13)(2,8,9,21)(4,12,16,26)(5,11,17,25)(7,19,18,29)(10,22,23,31)(14,24,27,32)(15,20,28,30)Polycyclic presentation:
2^8|b^2=g,d^2=h,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,e^b=eg,f^b=fh,e^c=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | #246930, #246932, #246939, … |
| maximal normal subgroup of | #246930, #246932, #246939, … |
| maximal quotient of | #165905, #165943, #165999, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11142);