Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18346 |
| 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 | #3490 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
536 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 536 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:
(1,2,5,10)(3,11,12,25)(4,15,18,9)(6,8,19,24)(7,16,22,13)(14,26,28,27)(17,30,32,23)(20,29,21,31);(1,3)(2,7)(4,16)(5,12)(6,14)(8,21)(9,25)(10,22)(11,15)(13,18)(17,31)(19,28)(20,24)(23,27)(26,30)(29,32);(4,17)(9,23)(11,26)(13,29)(15,30)(16,31)(18,32)(25,27);(2,8)(3,12)(7,20)(9,23)(10,24)(11,27)(13,16)(14,28)(15,30)(21,22)(25,26)(29,31)Polycyclic presentation:
2^8|a^2=g,e^2=g,b^a=be,c^a=cf,d^a=dh,e^a=eg,d^b=dg,e^b=eg,f^b=fh,e^c=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | #485085, #485086, #485103, … |
| maximal normal subgroup of | #485085, #485086, #485103, … |
| maximal quotient of | #168910, #168922, #168966, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11335);