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 | #2540 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
821 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,4,7,6)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30);(3,7)(4,6)(9,10,12,14)(11,15,16,13)(18,22)(23,25)(24,29)(30,31);(1,3,5,7)(2,6,8,4)(9,10)(11,13)(12,14)(15,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32);(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31);(1,4,5,6)(2,7,8,3)(9,11,12,16)(10,13,14,15)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)Polycyclic presentation:
2^8|a^2=g,b^2=gh,c^2=fg,d^2=f,e^2=h,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 | #6548626, #6548629, #6548638, … |
| maximal normal subgroup of | #6548626, #6548629, #6548638, … |
| maximal quotient of | #521630, #522258, #522901, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32185);