Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| 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 | |
771 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(11,15)(13,16)(17,18)(19,23)(20,27)(21,22)(24,28)(25,31)(26,29)(30,32);(2,6)(4,7)(9,10,12,14)(11,13,15,16)(18,22)(23,29)(24,27)(30,31);(1,2)(3,7)(4,8)(5,6)(9,11)(10,13)(12,15)(14,16)(17,19,21,26)(18,23,22,29)(20,25,28,32)(24,30,27,31);(1,3,5,8)(2,7,6,4)(17,20)(18,24)(19,25)(21,28)(22,27)(23,30)(26,32)(29,31);(1,4,5,7)(2,8,6,3)(17,18)(19,23)(20,24)(21,22)(25,30)(26,29)(27,28)(31,32)Polycyclic presentation:
2^8|b^2=g,c^2=f,d^2=h,e^2=h,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#39894, #7637551, #7638121, … |
| maximal normal subgroup of | C2×#39894, #7637551, #7638121, … |
| maximal quotient of | #521920, #522468, #522971, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32882);