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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
742 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(9,10,12,14)(11,13,15,16)(17,18,22,21)(19,25,27,23)(20,28,29,24)(26,30,32,31);(2,6)(4,7)(18,21)(23,25)(24,28)(30,31);(1,2)(3,7)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,19,22,27)(18,23,21,25)(20,26,29,32)(24,30,28,31);(1,3,5,8)(2,7,6,4)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32);(1,4,5,7)(2,8,6,3)(9,11,12,15)(10,13,14,16)(17,21,22,18)(19,25,27,23)(20,28,29,24)(26,31,32,30)Polycyclic presentation:
2^8|a^2=g,c^2=f,d^2=h,e^2=gh,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×#40264, #7637551, #7638134, … |
| maximal normal subgroup of | C2×#40264, #7637551, #7638134, … |
| maximal quotient of | #521945, #522488, #523049, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 33252);