Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
645 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,3)(4,7,11,9)(5,10,13,8)(12,15,16,14)(17,18)(19,24)(20,25)(21,29)(22,23)(26,30)(27,28)(31,32)(34,38)(35,41)(36,44)(46,48);(2,3)(4,11)(5,13)(14,15)(18,23)(20,24)(26,29)(28,31)(33,34,37,38)(35,42,41,39)(36,40,44,45)(43,48,47,46);(1,3,6,2)(4,7,11,9)(5,10,13,8)(12,14,16,15)(17,19,22,25)(18,24,23,20)(21,27,30,32)(26,31,29,28)(33,34)(35,39)(36,40)(37,38)(41,42)(43,46)(44,45)(47,48);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,20,22,24)(18,25,23,19)(21,28,30,31)(26,32,29,27)(33,35,37,41)(34,39,38,42)(36,43,44,47)(40,46,45,48);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,21)(18,26)(19,27)(20,28)(22,30)(23,29)(24,31)(25,32)(33,36,37,44)(34,40,38,45)(35,43,41,47)(39,46,42,48)Polycyclic presentation:
2^8|a^2=gh,b^2=g,c^2=fg,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#54582, #8103729, #8103904, … |
| maximal normal subgroup of | C2×#54582, #8103729, #8103904, … |
| maximal quotient of | #586274, #586941, #586970, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 47570);