Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
818 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,7)(3,8)(10,14)(15,16)(17,18,22,21)(19,23,25,27)(20,24,30,29)(26,31,28,32);(1,2,6,7)(3,4,8,5)(9,10)(11,15)(12,16)(13,14)(18,21)(19,25)(24,29)(26,28);(1,3)(2,4)(5,7)(6,8)(17,19)(18,23)(20,28)(21,27)(22,25)(24,32)(26,30)(29,31);(1,4,6,5)(2,8,7,3)(9,11,13,12)(10,15,14,16)(17,20,22,30)(18,24,21,29)(19,26,25,28)(23,31,27,32);(1,5,6,4)(2,3,7,8)(9,12,13,11)(10,16,14,15)(17,21,22,18)(19,27,25,23)(20,29,30,24)(26,32,28,31)Polycyclic presentation:
2^8|a^2=h,b^2=g,d^2=f,e^2=f,b^a=bf,c^a=cg,c^b=ch,e^b=eh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#35461, #7628831, #7628865, … |
| maximal normal subgroup of | C2×#35461, #7628831, #7628865, … |
| maximal quotient of | #515321, #515344, #515480, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28449);