Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18414 |
| 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 | #3561 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
454 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 454 conjugacy classes of subgroups — too many to draw. Only its 113 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 — 4 generators on 32 points:
(2,6)(4,11)(7,13)(8,12)(10,15)(14,16)(17,18,22,26)(19,27,29,32)(20,25,21,24)(23,30,31,28);(1,2,5,8)(3,7,9,14)(4,12,11,6)(10,16,15,13)(17,19,22,29)(18,23,26,31)(20,28,21,30)(24,27,25,32);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,15)(12,16)(20,21)(24,25)(27,32)(28,30);(3,9)(7,14)(10,15)(13,16)(17,20,22,21)(18,24,26,25)(19,28,29,30)(23,27,31,32)Polycyclic presentation:
2^8|a^2=h,b^2=g,d^2=h,e^2=g,b^a=be,c^a=cf,d^a=dh,e^a=eg,e^b=eg,f^b=fh,d^c=dg,e^c=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | #250428, #250430, #250431, … |
| maximal normal subgroup of | #250428, #250430, #250431, … |
| maximal quotient of | #169656, #169670, #169719, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11426);