Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| 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 | |
761 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)(3,5)(4,7)(6,8)(11,15)(13,16)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,30)(31,32);(1,3,4,8)(2,5,7,6)(9,10,12,14)(11,13,15,16)(18,22)(20,29)(23,25)(26,32);(1,3)(2,6)(4,8)(5,7)(9,11,12,15)(10,13,14,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32);(1,3)(2,6)(4,8)(5,7)(17,19,21,27)(18,23,22,25)(20,26,29,32)(24,30,28,31);(1,2)(3,6)(4,7)(5,8)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32)Polycyclic presentation:
2^8|b^2=g,c^2=fgh,d^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#45804, #7702392, #7702506, … |
| maximal normal subgroup of | C2×#45804, #7702392, #7702506, … |
| maximal quotient of | #557068, #557106, #557349, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38792);