Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | Aut(G) order 131072 |
| 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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
783 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)(3,8)(10,13)(14,16)(17,18,22,20)(19,23,25,26)(21,28,29,24)(27,32,31,30);(1,2,5,6)(3,7,8,4)(9,10)(11,14)(12,13)(15,16)(18,20)(19,25)(24,28)(27,31);(1,3)(2,7)(4,6)(5,8)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31);(1,4,5,7)(2,3,6,8)(9,11,12,15)(10,14,13,16)(17,20,22,18)(19,26,25,23)(21,28,29,24)(27,32,31,30);(17,21,22,29)(18,24,20,28)(19,27,25,31)(23,30,26,32)Polycyclic presentation:
2^8|a^2=h,b^2=g,d^2=f,e^2=h,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #129847, #129848, #163801, … |
| maximal normal subgroup of | #129847, #129848, #163801, … |
| maximal quotient of | #516414, #516629, #516645, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28302);