Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
823 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,22)(20,24)(21,23)(25,28)(26,29)(27,30)(31,32);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19)(18,22)(20,26)(21,25)(23,28)(24,29)(27,31)(30,32);(10,14)(13,16)(19,25)(22,28)(26,31)(29,32);(1,3,4,7)(2,6,5,8)(18,23)(22,28)(24,30)(29,32);(17,20,21,27)(18,24,23,30)(19,26,25,31)(22,29,28,32)Polycyclic presentation:
2^8|b^2=g,d^2=f,e^2=h,b^a=bf,c^a=cg,d^a=dh,c^b=ch64 conjugacy classes — view the full character table.
| maximal subgroup of | #129836, #251806, #251807, … |
| maximal normal subgroup of | #129836, #251806, #251807, … |
| maximal quotient of | #515045, #515062, #515074, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27628);