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:
(1,2,5,6)(3,9,10,7)(4,12,13,8)(11,14,16,15)(17,18)(19,25)(20,27)(21,29)(22,23)(24,30)(26,28)(31,32)(33,34,39,37)(35,42,36,40)(38,45,46,41)(43,47,44,48);(2,6)(3,10)(8,12)(11,16)(18,23)(20,25)(21,30)(26,32)(33,35,39,36)(34,40,37,42)(38,43,46,44)(41,47,45,48);(1,3,5,10)(2,7,6,9)(4,11,13,16)(8,14,12,15)(17,19,22,27)(18,20,23,25)(21,26,30,32)(24,28,29,31)(33,36,39,35)(34,40,37,42)(38,44,46,43)(41,47,45,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,20)(18,19)(21,28)(22,25)(23,27)(24,26)(29,32)(30,31)(33,37,39,34)(35,40,36,42)(38,45,46,41)(43,47,44,48);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,21,22,30)(18,24,23,29)(19,26,27,32)(20,28,25,31)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)Polycyclic presentation:
2^8|a^2=fgh,b^2=fg,c^2=f,d^2=fg,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×#54693, #8099791, #8100309, … |
| maximal normal subgroup of | C2×#54693, #8099791, #8100309, … |
| maximal quotient of | #586410, #586751, #586754, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 47681);