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 | |
740 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(10,14)(15,16)(18,19)(20,27)(24,26)(28,32);(2,5)(3,7)(17,18,22,19)(20,23,27,25)(21,26,29,24)(28,31,32,30);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,19,22,18)(20,25,27,23)(21,26,29,24)(28,31,32,30);(9,11,13,12)(10,15,14,16)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,14,15)(17,21,22,29)(18,24,19,26)(20,28,27,32)(23,30,25,31)Polycyclic presentation:
2^8|b^2=f,c^2=f,d^2=fg,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×#39098, #7636465, #7636536, … |
| maximal normal subgroup of | C2×#39098, #7636465, #7636536, … |
| maximal quotient of | #521581, #522279, #522832, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32086);