Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23998 |
| automorphism | Aut(G) = C2×#72236986 (order 4096) · Inn(G) = #2404 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2404 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3019 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
585 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 585 conjugacy classes of subgroups — too many to draw. Only its 121 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)(3,7)(4,14)(5,18)(6,10)(8,17)(9,20)(11,26)(12,28)(13,23)(15,31)(16,25)(19,24)(21,27)(22,30)(29,32);(1,3,6,13)(2,7,10,23)(4,15)(5,12,20,30)(8,21)(9,22,18,28)(11,16)(14,26)(17,29)(19,27)(24,32)(25,31);(1,4)(2,8)(3,11)(5,19)(6,16)(7,21)(9,25)(10,24)(12,29)(13,15)(14,18)(17,20)(22,31)(23,32)(26,28)(27,30);(5,20)(9,18)(12,30)(14,25)(17,19)(22,28)(26,31)(27,29)Polycyclic presentation:
2^8|b^2=eh,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,g^b=gh,e^c=eh,f^c=fh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #152685, #511854, #511856, … |
| maximal normal subgroup of | #152685, #511854, #511856, … |
| maximal quotient of | #251940, #251942, #251944, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16988);