Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | Aut(G) order 65536 |
| 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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
719 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(3,7)(6,8)(9,10,13,16)(11,14,12,15)(17,18)(19,20)(21,29)(22,23)(24,28)(25,30)(26,32)(27,31);(2,5)(3,7)(18,23)(19,24)(21,30)(27,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,19,22,24)(18,20,23,28)(21,26,30,31)(25,27,29,32);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,14,16,15)(17,20)(18,24)(19,23)(21,27)(22,28)(25,31)(26,29)(30,32);(1,2)(3,6)(4,5)(7,8)(9,12,13,11)(10,15,16,14)(17,21)(18,25)(19,26)(20,27)(22,30)(23,29)(24,31)(28,32)Polycyclic presentation:
2^8|a^2=g,c^2=f,d^2=fgh,e^2=g,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 | #7221572, #7221576, #7221636, … |
| maximal normal subgroup of | #7221572, #7221576, #7221636, … |
| maximal quotient of | #557279, #557343, #557484, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39104);