Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24472 |
| automorphism | Aut(G) = C2×#72236189 (order 2048) · Inn(G) = #2410 · Out(G) = C24 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2410 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3586 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
649 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,5,24,10)(3,12,16,14,17,26,15,8)(4,18,22,20,23,28,21,9)(13,25,30,32,31,29,19,27);(1,3)(2,8)(4,19)(5,14)(6,16)(7,17)(9,27)(10,26)(11,12)(13,22)(15,24)(18,25)(20,32)(21,31)(23,30)(28,29);(1,4,7,23)(2,9,5,20)(3,13,17,31)(6,21,24,22)(8,25,14,29)(10,18,11,28)(12,27,26,32)(15,30,16,19);(1,5)(2,7)(3,14)(4,20)(6,10)(8,17)(9,23)(11,24)(12,15)(13,29)(16,26)(18,21)(19,32)(22,28)(25,31)(27,30)Polycyclic presentation:
2^8|a^2=f,c^2=h,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,g^b=gh,f^c=fh,g^c=gh,e^d=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24474, #6437949, #6437951, … |
| maximal normal subgroup of | C2×#24474, #6437949, #6437951, … |
| maximal quotient of | #268287, #268289, #268292, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17462);