Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | Aut(G) order 196608 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
758 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,5,7,8)(10,14)(15,16)(19,25)(23,26)(27,31)(30,32);(1,2)(3,5)(4,6)(7,8)(17,18,22,20)(19,26,25,23)(21,24,29,28)(27,32,31,30);(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,19,22,25)(18,23,20,26)(21,27,29,31)(24,30,28,32);(9,11,13,12)(10,15,14,16)(17,20,22,18)(19,26,25,23)(21,24,29,28)(27,30,31,32);(1,3,4,7)(2,5,6,8)(9,12,13,11)(10,16,14,15)(17,21,22,29)(18,24,20,28)(19,27,25,31)(23,30,26,32)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=h,d^2=fg,e^2=g,b^a=bf,c^a=cg,c^b=ch,e^d=eh64 conjugacy classes — view the full character table.
| Fitting subgroup of | #36090⋊C3 |
| maximal subgroup of | #132442, #253628, #253630, … |
| maximal normal subgroup of | #132442, #253628, #253630, … |
| maximal quotient of | #517615, #517620, #517625, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29078);