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 | |
643 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(3,9)(8,13)(11,15)(17,18,22,23)(19,20,24,27)(21,29,30,25)(26,32,31,28)(34,38)(35,41)(36,44)(46,48);(4,12)(8,13)(11,15)(14,16)(18,23)(20,27)(25,29)(28,32)(33,34,37,38)(35,42,41,39)(36,40,44,45)(43,48,47,46);(1,2,5,6)(3,10,9,7)(4,8,12,13)(11,16,15,14)(17,19,22,24)(18,20,23,27)(21,26,30,31)(25,28,29,32)(33,34)(35,39)(36,40)(37,38)(41,42)(43,46)(44,45)(47,48);(1,3,5,9)(2,7,6,10)(4,11,12,15)(8,14,13,16)(17,20)(18,24)(19,23)(21,28)(22,27)(25,31)(26,29)(30,32)(33,35,37,41)(34,39,38,42)(36,43,44,47)(40,46,45,48);(1,4)(2,8)(3,11)(5,12)(6,13)(7,14)(9,15)(10,16)(17,21,22,30)(18,25,23,29)(19,26,24,31)(20,28,27,32)(33,36,37,44)(34,40,38,45)(35,43,41,47)(39,46,42,48)Polycyclic presentation:
2^8|a^2=fh,b^2=h,c^2=fg,d^2=g,e^2=f,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×#55847, #8101701, #8101958, … |
| maximal normal subgroup of | C2×#55847, #8101701, #8101958, … |
| maximal quotient of | #585945, #586459, #586760, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 48835);