Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| 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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
752 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)(4,7)(10,13)(14,16)(17,18,23,26)(19,21,20,24)(22,30,31,25)(27,32,28,29);(2,6)(3,8)(17,19,23,20)(18,24,26,21)(22,27,31,28)(25,32,30,29);(1,2,5,6)(3,7,8,4)(9,10)(11,14)(12,13)(15,16)(17,20,23,19)(18,24,26,21)(22,28,31,27)(25,32,30,29);(1,3)(2,7)(4,6)(5,8)(17,21,23,24)(18,19,26,20)(22,29,31,32)(25,27,30,28);(1,4,5,7)(2,3,6,8)(9,11,12,15)(10,14,13,16)(17,22,23,31)(18,25,26,30)(19,27,20,28)(21,29,24,32)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=fh,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#37056, #7634359, #7634415, … |
| maximal normal subgroup of | C2×#37056, #7634359, #7634415, … |
| maximal quotient of | #553907, #553910, #554116, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30044);