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 | |
615 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(17,18,22,20)(19,23,25,27)(21,28,29,24)(26,32,31,30)(33,34)(35,37)(36,43)(38,40)(39,41)(42,45)(44,47)(46,48);(2,6)(4,12)(7,9)(10,16)(18,20)(19,25)(21,29)(30,32)(33,35,39,36)(34,37,41,43)(38,44,46,42)(40,47,48,45);(1,2,5,6)(3,9,11,7)(4,8,12,13)(10,15,16,14)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(33,36,39,35)(34,37,41,43)(38,44,46,42)(40,45,48,47);(1,3,5,11)(2,7,6,9)(4,10,12,16)(8,14,13,15)(17,20,22,18)(19,23,25,27)(21,28,29,24)(26,30,31,32)(33,37)(34,36)(35,41)(38,45)(39,43)(40,44)(42,48)(46,47);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,21,22,29)(18,24,20,28)(19,26,25,31)(23,30,27,32)(33,38,39,46)(34,40,41,48)(35,42,36,44)(37,45,43,47)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=fgh,d^2=h,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×#58670, #8104799, #8104903, … |
| maximal normal subgroup of | C2×#58670, #8104799, #8104903, … |
| maximal quotient of | #586358, #586533, #587168, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51658);