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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
749 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,4)(3,7)(9,10,12,14)(11,15,16,13)(20,28)(24,29)(26,32)(30,31);(1,2,5,4)(3,8,7,6)(11,16)(13,15)(18,22)(23,25)(24,29)(30,31);(3,7)(6,8)(9,11)(10,13)(12,16)(14,15)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30);(1,3)(2,6)(4,8)(5,7)(9,10)(11,13)(12,14)(15,16)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31);(1,4,5,2)(3,8,7,6)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=gh,d^2=gh,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×#52928, #8092758, #8092766, … |
| maximal normal subgroup of | C2×#52928, #8092758, #8092766, … |
| maximal quotient of | #582186, #582427, #582428, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45916);