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 | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
816 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(3,7)(6,8)(10,13)(11,15)(17,18,22,25)(19,23,20,27)(21,29,30,24)(26,32,28,31);(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,13)(14,15)(17,19,22,20)(18,23,25,27)(21,26,30,28)(24,31,29,32);(2,5)(6,8)(11,15)(14,16)(17,20,22,19)(18,23,25,27)(21,28,30,26)(24,31,29,32);(1,2)(3,6)(4,5)(7,8)(9,11,12,15)(10,14,13,16)(17,18)(19,23)(20,27)(21,24)(22,25)(26,31)(28,32)(29,30);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,29)(27,32)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=g,d^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×#52449, #8092171, #8092253, … |
| maximal normal subgroup of | C2×#52449, #8092171, #8092253, … |
| maximal quotient of | #582266, #582443, #582452, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45437);