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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
772 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(9,10,13,11)(12,15,16,14)(17,18,22,23)(19,20,24,28)(21,29,30,25)(26,32,31,27);(1,2)(3,7)(4,8)(5,6)(9,11,13,10)(12,15,16,14)(18,23)(19,24)(21,30)(27,32);(2,6)(3,8)(9,12)(10,14)(11,15)(13,16)(17,19)(18,20)(21,26)(22,24)(23,28)(25,27)(29,32)(30,31);(1,3)(2,4)(5,8)(6,7)(17,20)(18,24)(19,23)(21,27)(22,28)(25,31)(26,29)(30,32);(1,4)(2,3)(5,7)(6,8)(17,21,22,30)(18,25,23,29)(19,26,24,31)(20,27,28,32)Polycyclic presentation:
2^8|a^2=fg,b^2=g,e^2=f,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×#47912, #7703665, #7704095, … |
| maximal normal subgroup of | C2×#47912, #7703665, #7704095, … |
| maximal quotient of | #560450, #560570, #561457, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 40900);