Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| 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 | |
793 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,8)(3,5,7,4)(9,10,13,16)(11,14,12,15)(20,27)(24,29)(26,31)(30,32);(1,3,6,7)(2,4,8,5)(9,10)(11,14)(12,15)(13,16)(18,22)(19,25)(24,29)(26,31);(1,2)(3,4)(5,7)(6,8)(9,11,13,12)(10,14,16,15)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32);(1,4,6,5)(2,3,8,7)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32);(1,5,6,4)(2,7,8,3)(9,12,13,11)(10,15,16,14)(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=fgh,d^2=g,e^2=f,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#35128, #7628338, #7628361, … |
| maximal normal subgroup of | C2×#35128, #7628338, #7628361, … |
| maximal quotient of | #516521, #516555, #516688, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28116);