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 |
| prime radical | #3531 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
405 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18,23,26)(19,29,32,41)(20,33,35,42)(21,28,22,27)(24,36,39,31)(25,37,40,34)(30,38,46,48)(43,44,47,45);(1,2)(3,7)(4,8)(5,13)(6,9)(10,14)(11,15)(12,16)(17,19,23,32)(18,24,26,39)(20,30,35,46)(21,36,22,31)(25,38,40,48)(27,41,28,29)(33,47,42,43)(34,45,37,44);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,20)(18,25)(19,30)(21,34)(22,37)(23,35)(24,38)(26,40)(27,33)(28,42)(29,43)(31,44)(32,46)(36,45)(39,48)(41,47);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(18,26)(24,39)(25,40)(27,28)(29,41)(33,42)(38,48)(43,47)Polycyclic presentation:
2^8|a^2=g,b^2=g,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×#19013, #6351398, #6351400, … |
| maximal normal subgroup of | C2×#19013, #6351398, #6351400, … |
| maximal quotient of | #165906, #165908, #165944, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12001);