Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| 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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
763 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,4)(3,7,5,8)(9,10,13,11)(12,15,16,14)(17,18,22,20)(19,23,25,27)(21,24,29,28)(26,30,31,32);(2,4)(3,5)(9,11,13,10)(12,15,16,14)(19,25)(21,29)(23,27)(24,28);(1,3,6,5)(2,7,4,8)(9,12)(10,14)(11,15)(13,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31);(1,4,6,2)(3,7,5,8)(17,20,22,18)(19,23,25,27)(21,28,29,24)(26,30,31,32);(1,5,6,3)(2,7,4,8)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)Polycyclic presentation:
2^8|a^2=gh,b^2=g,c^2=f,d^2=h,e^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#47764, #7701750, #7701761, … |
| maximal normal subgroup of | C2×#47764, #7701750, #7701761, … |
| maximal quotient of | #576693, #576724, #576767, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 40752);