Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| automorphism | Aut(G) order 262144 |
| 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 | |
791 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(6,8)(9,10,13,11)(12,15,16,14)(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,30,32,31);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,15,16,14)(18,19)(21,29)(23,25)(27,32);(1,3,4,7)(2,6,5,8)(9,12)(10,14)(11,15)(13,16)(17,19,22,18)(20,23,28,25)(21,26,29,24)(27,30,32,31);(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31);(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)Polycyclic presentation:
2^8|a^2=gh,b^2=g,c^2=f,d^2=h,b^a=bf,c^a=cg,c^b=ch,e^b=eh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #97480, #129442, #134180, … |
| maximal normal subgroup of | #97480, #129442, #134180, … |
| maximal quotient of | #515304, #515334, #515370, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28615);