Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31414 |
| automorphism | Aut(G) = C2×#72236189 (order 2048) · Inn(G) = #2688 · Out(G) = C24 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2688 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
333 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 333 conjugacy classes of subgroups — too many to draw. Only its 97 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)(3,13,16,8)(4,18,21,30)(5,11,6,10)(9,23,28,20)(14,27,32,25)(15,31,19,24)(17,26,22,29);(1,3,5,22,7,16,6,17)(2,8,10,29,12,13,11,26)(4,19,20,32,21,15,23,14)(9,27,18,31,28,25,30,24);(1,4)(2,9)(3,14,16,32)(5,23)(6,20)(7,21)(8,24,13,31)(10,30)(11,18)(12,28)(15,17,19,22)(25,26,27,29);(1,4)(2,9)(3,15)(5,23)(6,20)(7,21)(8,25)(10,30)(11,18)(12,28)(13,27)(14,17)(16,19)(22,32)(24,26)(29,31)Polycyclic presentation:
2^8|a^2=h,b^2=ef,c^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=def,f^a=fh,g^a=gh,c^b=cg,d^b=de,e^b=eh,f^b=fh,f^c=fh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #6792410, #6792413, #6792417, … |
| maximal normal subgroup of | #6792410, #6792413, #6792417, … |
| maximal quotient of | #346000, #346002, #346005, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24404);