Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| 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 | C22×#41 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
852 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(6,8)(17,18)(19,23)(20,27)(21,22)(24,28)(25,31)(26,29)(30,32);(2,5)(3,7)(10,13)(11,15)(18,22)(23,29)(24,27)(30,31);(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,26)(18,23,22,29)(20,25,28,32)(24,30,27,31);(17,20,21,28)(18,24,22,27)(19,25,26,32)(23,30,29,31);(1,3)(2,6)(4,7)(5,8)(9,11,12,15)(10,14,13,16)(17,18)(19,23)(20,24)(21,22)(25,30)(26,29)(27,28)(31,32)Polycyclic presentation:
2^8|c^2=fg,d^2=f,e^2=gh,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh58 conjugacy classes — view the full character table.
| maximal subgroup of | #6534614, #6534616, #6534620, … |
| maximal normal subgroup of | #6534614, #6534616, #6534620, … |
| maximal quotient of | #521546, #522243, #522795, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32056);