Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23994 |
| automorphism | Aut(G) = C2×#72237558 (order 4096) · Inn(G) = #535 · Out(G) = C26 |
| 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 | #2487 |
| prime radical | #532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
317 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 317 conjugacy classes of subgroups — too many to draw. Only its 117 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:
(1,2,7,12,8,14,6,13)(3,15,25,29,18,32,17,9)(4,19,26,30,22,28,21,10)(5,23,16,27,24,31,20,11);(1,3,6,17,8,18,7,25)(2,9,12,15,14,29,13,32)(4,20,21,24,22,16,26,5)(10,27,19,31,30,11,28,23);(1,4)(2,10)(3,16)(5,17)(6,21)(7,26)(8,22)(9,27)(11,29)(12,19)(13,28)(14,30)(15,31)(18,20)(23,32)(24,25);(1,5,6,20,8,24,7,16)(2,11,12,23,14,27,13,31)(3,4,17,21,18,22,25,26)(9,28,15,10,29,19,32,30)Polycyclic presentation:
2^8|a^2=eg,b^2=e,d^2=e,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,g^a=gh,c^b=cg,f^b=fh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #139579, #139580, #139581, … |
| maximal normal subgroup of | #139579, #139580, #139581, … |
| maximal quotient of | #314957, #314958, #314959, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20681);