Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18162 |
| 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 | #3492 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
499 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 499 conjugacy classes of subgroups — too many to draw. Only its 117 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:
(1,2,8,18)(3,12,23,32)(4,6,11,7)(5,17,24,28)(9,25,31,15)(10,21,20,22)(13,29,26,19)(14,27,16,30);(1,3)(2,9)(4,13)(5,19)(6,16)(7,14)(8,23)(10,27)(11,26)(12,17)(15,22)(18,31)(20,30)(21,25)(24,29)(28,32);(1,4,7,2)(3,13,14,9)(5,20,22,28)(6,18,8,11)(10,21,17,24)(12,29,27,25)(15,32,19,30)(16,31,23,26);(3,14)(9,13)(12,27)(15,19)(16,23)(25,29)(26,31)(30,32)Polycyclic presentation:
2^8|a^2=gh,c^2=g,e^2=g,b^a=be,c^a=cf,e^a=eg,d^b=dg,e^b=eg,f^b=fh,e^c=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | #140043, #140045, #140062, … |
| maximal normal subgroup of | #140043, #140045, #140062, … |
| maximal quotient of | #166300, #166316, #166356, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11160);