Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| 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 | C22×#41 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
845 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,4,7,8)(9,10,13,12)(11,15,16,14)(19,25)(22,26)(27,31)(30,32);(2,6)(3,7)(10,12)(14,15)(17,18,21,24)(19,26,25,22)(20,28,29,23)(27,30,31,32);(9,11)(10,14)(12,15)(13,16)(17,18)(19,22)(20,23)(21,24)(25,26)(27,30)(28,29)(31,32);(1,3)(2,4)(5,7)(6,8)(9,12,13,10)(11,15,16,14)(17,19,21,25)(18,22,24,26)(20,27,29,31)(23,30,28,32);(1,4)(2,3)(5,8)(6,7)(9,11)(10,14)(12,15)(13,16)(17,20,21,29)(18,23,24,28)(19,27,25,31)(22,30,26,32)Polycyclic presentation:
2^8|a^2=f,b^2=h,d^2=gh,e^2=h,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #97168, #130091, #133462, … |
| maximal normal subgroup of | #97168, #130091, #133462, … |
| maximal quotient of | #557186, #557527, #557677, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39304);