Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| 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×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
793 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(7,8)(10,13)(14,16)(19,25)(23,26)(27,31)(30,32);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,22,20)(19,26,25,23)(21,24,29,28)(27,32,31,30);(1,2)(3,7)(4,8)(5,6)(17,19,22,25)(18,23,20,26)(21,27,29,31)(24,30,28,32);(1,3,5,4)(2,7,6,8)(17,20,22,18)(19,26,25,23)(21,24,29,28)(27,30,31,32);(1,4,5,3)(2,8,6,7)(9,11,12,15)(10,14,13,16)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)Polycyclic presentation:
2^8|b^2=h,c^2=h,d^2=g,e^2=f,b^a=bf,c^a=cg,c^b=ch,e^d=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | #132438, #253627, #253629, … |
| maximal normal subgroup of | #132438, #253627, #253629, … |
| maximal quotient of | #517610, #517611, #517620, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29069);