Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| 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 | |
647 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,9)(7,12)(10,15)(14,16)(17,18,21,24)(19,22,26,29)(20,23,27,30)(25,28,31,32)(34,38)(35,41)(36,44)(46,48);(2,6)(4,11)(7,12)(10,15)(20,27)(23,30)(25,31)(28,32)(33,34,37,38)(35,42,41,39)(36,40,44,45)(43,48,47,46);(1,2,5,6)(3,7,9,12)(4,8,11,13)(10,14,15,16)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32)(33,34)(35,39)(36,40)(37,38)(41,42)(43,46)(44,45)(47,48);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(17,19,21,26)(18,22,24,29)(20,25,27,31)(23,28,30,32)(33,35,37,41)(34,39,38,42)(36,43,44,47)(40,46,45,48);(1,4)(2,8)(3,10)(5,11)(6,13)(7,14)(9,15)(12,16)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32)(33,36,37,44)(34,40,38,45)(35,43,41,47)(39,46,42,48)Polycyclic presentation:
2^8|a^2=fg,b^2=f,c^2=fh,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=df,e^b=egh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#53331, #8105918, #8105927, … |
| maximal normal subgroup of | C2×#53331, #8105918, #8105927, … |
| maximal quotient of | #606830, #606844, #606849, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46319);