Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23995 |
| automorphism | Aut(G) = #72337551 (order 2048) · Inn(G) = #535 · Out(G) = C22×D4 |
| 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 | |
302 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 302 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)(5,20)(6,11)(7,14)(8,15)(9,18)(10,21)(13,27)(17,22)(19,30)(23,29)(24,31)(25,28)(26,32);(1,3)(2,7,11,23)(4,13)(5,14)(6,15)(8,10,12,20)(9,22,26,27)(16,25,30,24)(17,19)(18,28)(21,29)(31,32);(1,3)(2,8,11,12)(4,17)(5,14)(6,15)(7,20,23,10)(9,25,26,24)(13,19)(16,27,30,22)(18,31)(21,29)(28,32);(1,4,5,18,6,19,21,32)(2,9,10,16,11,26,20,30)(3,13,14,28,15,17,29,31)(7,22,12,25,23,27,8,24)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,d^c=dh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #500650, #500651, #500940, … |
| maximal normal subgroup of | #500650, #500651, #500940, … |
| maximal quotient of | #326560, #326561, #326562, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22259);