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, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
617 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(3,9)(8,13)(11,15)(17,18)(19,24)(20,25)(21,28)(22,23)(26,31)(27,30)(29,32)(33,34,39,41)(35,42,36,37)(38,46,47,40)(43,45,44,48);(4,12)(8,13)(11,15)(14,16)(18,23)(19,27)(20,29)(21,31)(33,35,39,36)(34,37,41,42)(38,43,47,44)(40,45,46,48);(1,2,5,6)(3,10,9,7)(4,8,12,13)(11,16,15,14)(17,19,22,27)(18,24,23,30)(20,26,29,28)(21,25,31,32)(33,36,39,35)(34,37,41,42)(38,44,47,43)(40,45,46,48);(1,3,5,9)(2,7,6,10)(4,11,12,15)(8,14,13,16)(17,20,22,29)(18,25,23,32)(19,28,27,26)(21,30,31,24)(33,37)(34,35)(36,41)(38,45)(39,42)(40,43)(44,46)(47,48);(1,4,5,12)(2,8,6,13)(3,11,9,15)(7,14,10,16)(17,21)(18,26)(19,25)(20,30)(22,31)(23,28)(24,29)(27,32)(33,38)(34,40)(35,43)(36,44)(37,45)(39,47)(41,46)(42,48)Polycyclic presentation:
2^8|a^2=fh,b^2=fh,c^2=fg,d^2=gh,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×#58550, #8104419, #8104426, … |
| maximal normal subgroup of | C2×#58550, #8104419, #8104426, … |
| maximal quotient of | #585910, #586116, #587068, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51538);