Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18316 |
| automorphism | Aut(G) order 4096 |
| 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 | #3459 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
504 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 504 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) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11)(3,13,16,28)(4,10,5,19)(7,12,23,29)(8,20,26,14)(9,18,21,22)(15,31,17,24)(25,30,27,32);(1,3)(2,8)(4,14)(5,20)(6,16)(7,17)(9,24)(10,28)(11,26)(12,27)(13,19)(15,23)(18,30)(21,31)(22,32)(25,29);(1,2)(3,8)(4,9)(5,21)(6,11)(7,12)(10,22)(13,30)(14,24)(15,25)(16,26)(17,27)(18,19)(20,31)(23,29)(28,32);(1,4,6,5)(2,9,11,21)(3,14,16,20)(7,18,23,22)(8,24,26,31)(10,12,19,29)(13,25,28,27)(15,32,17,30)Polycyclic presentation:
2^8|a^2=g,d^2=g,e^2=g,b^a=be,c^a=cf,d^a=dgh,e^a=eg,e^b=eg,f^b=fh,e^c=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | #485953, #485954, #485959, … |
| maximal normal subgroup of | #485953, #485954, #485959, … |
| maximal quotient of | #168506, #168535, #168567, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11311);