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:
(1,2)(3,7)(4,12)(5,6)(8,13)(9,10)(11,16)(14,15)(17,18,22,23)(19,20,24,27)(21,29,30,25)(26,32,31,28)(33,34,39,42)(35,40,36,37)(38,41,46,48)(43,47,44,45);(2,6)(3,9)(4,13)(14,16)(18,23)(20,27)(25,29)(28,32)(33,35,39,36)(34,40,42,37)(38,44,46,43)(41,45,48,47);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,19,22,24)(18,20,23,27)(21,26,30,31)(25,28,29,32)(33,36,39,35)(34,40,42,37)(38,44,46,43)(41,47,48,45);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,16)(13,15)(17,20)(18,24)(19,23)(21,28)(22,27)(25,31)(26,29)(30,32)(33,37,39,40)(34,36,42,35)(38,45,46,47)(41,44,48,43);(1,4,5,13)(2,8,6,12)(3,11,9,15)(7,14,10,16)(17,21,22,30)(18,25,23,29)(19,26,24,31)(20,28,27,32)(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=g,c^2=fgh,d^2=g,e^2=f,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×#56826, #8100074, #8100300, … |
| maximal normal subgroup of | C2×#56826, #8100074, #8100300, … |
| maximal quotient of | #586237, #587167, #587280, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49814);