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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
719 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)(7,8)(9,10,13,12)(11,14,16,15)(18,20)(19,25)(24,28)(27,31);(9,11)(10,14)(12,15)(13,16)(17,18,22,20)(19,23,25,26)(21,28,29,24)(27,32,31,30);(1,2)(3,7)(4,8)(5,6)(9,10,13,12)(11,15,16,14)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31);(1,3,5,4)(2,7,6,8)(17,20,22,18)(19,26,25,23)(21,28,29,24)(27,32,31,30);(1,4,5,3)(2,8,6,7)(9,12,13,10)(11,14,16,15)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=h,d^2=g,e^2=fgh,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×#39579, #7636830, #7636858, … |
| maximal normal subgroup of | C2×#39579, #7636830, #7636858, … |
| maximal quotient of | #521462, #522224, #522622, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32567);