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, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
699 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(7,8)(17,18,22,21)(19,23,27,26)(20,28,29,24)(25,31,32,30);(9,10,13,16)(11,15,12,14)(18,21)(23,26)(24,28)(30,31);(1,2)(3,7)(4,8)(5,6)(9,11,13,12)(10,14,16,15)(17,19,22,27)(18,23,21,26)(20,25,29,32)(24,30,28,31);(1,3,5,4)(2,7,6,8)(17,20,22,29)(18,24,21,28)(19,25,27,32)(23,30,26,31);(1,4,5,3)(2,8,6,7)(9,12,13,11)(10,15,16,14)(17,21,22,18)(19,26,27,23)(20,28,29,24)(25,31,32,30)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=fh,d^2=fg,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×#39715, #7637186, #7637289, … |
| maximal normal subgroup of | C2×#39715, #7637186, #7637289, … |
| maximal quotient of | #521520, #522219, #522765, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32703);