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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
573 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 573 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)(3,11)(4,8)(5,18)(6,10)(7,15)(9,21)(12,19)(13,29)(14,25)(16,26)(17,24)(20,23)(22,31)(27,30)(28,32);(1,2)(3,12,14,7)(4,8)(5,9)(6,10)(11,15,25,19)(13,28,30,22)(16,23)(17,24)(18,21)(20,26)(27,32,29,31);(1,3)(2,7)(4,13)(5,19)(6,14)(8,22)(9,25)(10,12)(11,18)(15,21)(16,32)(17,30)(20,31)(23,27)(24,28)(26,29);(1,4,6,17)(2,8,10,24)(3,13,14,30)(5,20,21,16)(7,22,12,28)(9,26,18,23)(11,27,25,29)(15,32,19,31)Polycyclic presentation:
2^8|b^2=e,d^2=h,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 | #152682, #511859, #511860, … |
| maximal normal subgroup of | #152682, #511859, #511860, … |
| maximal quotient of | #251936, #251938, #251939, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16987);