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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
663 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)(17,18,22,21)(19,25,26,23)(20,28,29,24)(27,30,32,31)(33,34)(35,38)(36,44)(37,40)(39,45)(41,43)(42,47)(46,48);(4,12)(8,13)(11,15)(14,16)(19,26)(23,25)(27,32)(30,31)(33,34,37,40)(35,41,43,38)(36,44,45,39)(42,46,48,47);(1,2,5,6)(3,10,9,7)(4,8,12,13)(11,16,15,14)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,45)(46,48);(1,3,5,9)(2,7,6,10)(4,11,12,15)(8,14,13,16)(17,20,22,29)(18,24,21,28)(19,27,26,32)(23,30,25,31)(33,35,37,43)(34,38,40,41)(36,42,45,48)(39,46,44,47);(1,4,5,12)(2,8,6,13)(3,11,9,15)(7,14,10,16)(17,21,22,18)(19,25,26,23)(20,28,29,24)(27,31,32,30)(33,36,37,45)(34,39,40,44)(35,42,43,48)(38,46,41,47)Polycyclic presentation:
2^8|a^2=fh,b^2=f,c^2=fgh,d^2=fg,e^2=fg,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×#57093, #8099569, #8100304, … |
| maximal normal subgroup of | C2×#57093, #8099569, #8100304, … |
| maximal quotient of | #586455, #587135, #587286, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 50081);