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 | |
665 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)(19,22)(20,25)(21,26)(23,27)(24,29)(28,30)(31,32)(33,34,37,40)(35,41,43,38)(36,39,44,46)(42,47,48,45);(4,12)(8,13)(11,15)(14,16)(18,19)(20,27)(24,26)(28,32)(33,34,37,40)(35,38,43,41)(36,39,44,46)(42,45,48,47);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,19,22,18)(20,25,27,23)(21,26,29,24)(28,31,32,30)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,15)(13,16)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32)(33,35)(34,38)(36,42)(37,43)(39,45)(40,41)(44,48)(46,47);(1,4,5,12)(2,8,6,13)(3,11,9,15)(7,14,10,16)(17,21,22,29)(18,24,19,26)(20,28,27,32)(23,30,25,31)(33,36,37,44)(34,39,40,46)(35,42,43,48)(38,45,41,47)Polycyclic presentation:
2^8|a^2=fh,b^2=fh,c^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×#55940, #8099886, #8100307, … |
| maximal normal subgroup of | C2×#55940, #8099886, #8100307, … |
| maximal quotient of | #586641, #586971, #587283, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 48928);