Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23360 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2380 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | #532 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
344 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 344 conjugacy classes of subgroups — too many to draw. Only its 133 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)(3,11,14,7)(4,15,18,29)(5,9)(6,10)(8,17,25,31)(12,23)(13,26,19,22)(16,21)(20,24)(27,30)(28,32);(1,3)(2,7)(4,16)(5,19)(6,14)(8,23)(9,26)(10,11)(12,17)(13,20)(15,27)(18,30)(21,29)(22,24)(25,32)(28,31);(1,4,5,17,6,18,20,31)(2,8,24,29,10,25,9,15)(3,12,13,16,14,28,19,30)(7,21,26,23,11,27,22,32);(2,9,10,24)(7,22,11,26)(8,15,25,29)(21,32,27,23)Polycyclic presentation:
2^8|a^2=g,c^2=ef,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=def,e^a=eh,c^b=cef,d^b=de,f^b=fh,g^b=gh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #139424, #139426, #139428, … |
| maximal normal subgroup of | #139424, #139426, #139428, … |
| maximal quotient of | #232122, #232124, #232126, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16348);