Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18992 |
| automorphism | Aut(G) = C26×#120 (order 2048) · Inn(G) = C2×#120 · Out(G) = C25 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11)(4,10)(5,9)(6,8)(7,12)(13,16)(14,15)(18,23)(20,21)(24,31)(25,28)(26,29)(30,32);(1,3)(2,7)(4,14)(5,12)(6,13)(8,15)(9,16)(10,11)(17,18)(19,24)(20,23)(21,28)(22,25)(26,30)(27,32)(29,31);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,19)(18,24)(20,26)(21,29)(22,27)(23,30)(25,32)(28,31);(2,8)(7,15)(9,10)(11,16)(17,20,22,21)(18,23,25,28)(19,26,27,29)(24,30,32,31)Polycyclic presentation:
2^8|d^2=h,e^2=g,f^2=gh,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fgh,e^b=eg,f^b=fh,e^c=eh,f^c=fgh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#19010, #6351398, #6351399, … |
| maximal normal subgroup of | C2×#19010, #6351398, #6351399, … |
| maximal quotient of | #165903, #165909, #165941, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11998);