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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
768 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)(9,10,12,13)(11,14,15,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,31)(28,32);(2,5)(6,8)(17,19,21,26)(18,22,24,30)(20,27,28,25)(23,31,32,29);(10,13)(14,16)(17,20,21,28)(18,23,24,32)(19,25,26,27)(22,29,30,31);(9,11,12,15)(10,14,13,16)(18,24)(22,30)(23,32)(29,31);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=h,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dh,c^b=ch64 conjugacy classes — view the full character table.
| maximal subgroup of | #133254, #252253, #252254, … |
| maximal normal subgroup of | #133254, #252253, #252254, … |
| maximal quotient of | #515095, #515097, #515112, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27592);