Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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 | |
764 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,5)(6,8)(10,14)(15,16)(17,18,22,23)(19,20,24,28)(21,29,30,25)(26,32,31,27);(2,5)(6,7)(18,23)(19,24)(21,30)(27,32);(9,10,13,14)(11,15,12,16)(17,19)(18,20)(21,26)(22,24)(23,28)(25,27)(29,32)(30,31);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,15,14,16)(17,20)(18,24)(19,23)(21,27)(22,28)(25,31)(26,29)(30,32);(1,3,4,8)(2,6,5,7)(9,12,13,11)(10,16,14,15)(17,21,22,30)(18,25,23,29)(19,26,24,31)(20,27,28,32)Polycyclic presentation:
2^8|a^2=h,c^2=g,d^2=g,e^2=fg,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#49460, #7704557, #7704978, … |
| maximal normal subgroup of | C2×#49460, #7704557, #7704978, … |
| maximal quotient of | #560783, #561098, #561887, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 42448);