Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| 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 | |
753 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,3)(4,7,8,6)(9,10,14,15)(11,13,12,16)(17,18,23,20)(19,21,24,27)(22,25,31,28)(26,29,30,32);(2,3)(6,7)(10,15)(11,12)(17,19,23,24)(18,21,20,27)(22,30,31,26)(25,32,28,29);(9,11,14,12)(10,16,15,13)(17,20,23,18)(19,21,24,27)(22,28,31,25)(26,29,30,32);(1,3,5,2)(4,7,8,6)(9,12,14,11)(10,13,15,16)(17,21)(18,24)(19,20)(22,29)(23,27)(25,30)(26,28)(31,32);(1,4,5,8)(2,6,3,7)(9,13)(10,11)(12,15)(14,16)(17,22)(18,25)(19,26)(20,28)(21,29)(23,31)(24,30)(27,32)Polycyclic presentation:
2^8|a^2=fgh,b^2=gh,c^2=h,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#57787, #8101461, #8101964, … |
| maximal normal subgroup of | C2×#57787, #8101461, #8101964, … |
| maximal quotient of | #585839, #586900, #587311, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 50775);