Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23994 |
| automorphism | Aut(G) = C2×#72236881 (order 2048) · Inn(G) = #535 · Out(G) = C25 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #535 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | #2580 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
306 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 306 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)(3,12)(4,16)(5,19)(6,11)(7,14)(8,15)(9,17)(10,20)(13,26)(18,30)(21,28)(22,29)(23,32)(24,31)(25,27);(1,3)(2,7,11,22)(4,13)(5,14)(6,15)(8,10,12,19)(9,21,24,26)(16,25,30,23)(17,27)(18,28)(20,29)(31,32);(1,3)(2,8,11,12)(4,13)(5,14)(6,15)(7,19,22,10)(9,23,24,25)(16,21,30,26)(17,27)(18,28)(20,29)(31,32);(1,4,5,17,6,18,20,31)(2,9,10,16,11,24,19,30)(3,13,14,27,15,28,29,32)(7,21,12,25,22,26,8,23)Polycyclic presentation:
2^8|b^2=g,c^2=g,d^2=e,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gh,c^b=cg,f^b=fh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #500646, #500647, #500648, … |
| maximal normal subgroup of | #500646, #500647, #500648, … |
| maximal quotient of | #326551, #326552, #326553, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22258);