Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| 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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
631 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)(3,7,9,12)(4,8,11,13)(10,14,15,16)(17,18)(19,20)(21,25)(22,23)(24,27)(26,28)(29,30)(31,32)(36,44)(39,46)(42,48)(45,47);(2,6)(3,9)(8,13)(10,15)(18,23)(20,24)(25,30)(28,31)(33,34,37,40)(35,38,43,41)(36,39,44,46)(42,45,48,47);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,30)(29,32)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,46)(45,48);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,13)(15,16)(17,20,22,24)(18,19,23,27)(21,28,29,31)(25,26,30,32)(33,35)(34,38)(36,42)(37,43)(39,45)(40,41)(44,48)(46,47);(1,4,5,11)(2,8,6,13)(3,10,9,15)(7,14,12,16)(17,21,22,29)(18,25,23,30)(19,26,27,32)(20,28,24,31)(33,36)(34,39)(35,42)(37,44)(38,45)(40,46)(41,47)(43,48)Polycyclic presentation:
2^8|a^2=fg,b^2=fgh,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dfg,e^a=efgh,c^b=ch,d^b=df,d^c=dfgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#52786, #8092543, #8092689, … |
| maximal normal subgroup of | C2×#52786, #8092543, #8092689, … |
| maximal quotient of | #582316, #582513, #582561, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45774);