Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| 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 | |
808 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,7)(3,8,5,4)(9,10,13,12)(11,15,16,14)(18,22)(20,28)(23,25)(26,32);(2,7)(4,8)(18,22)(20,28)(23,25)(26,32);(1,3,6,5)(2,8,7,4)(9,11,13,16)(10,14,12,15)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32);(1,4)(2,3)(5,7)(6,8)(9,10,13,12)(11,14,16,15)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31);(1,5,6,3)(2,4,7,8)(9,12,13,10)(11,15,16,14)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)Polycyclic presentation:
2^8|a^2=fgh,c^2=fgh,d^2=g,e^2=fg,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#44739, #7668958, #7669114, … |
| maximal normal subgroup of | C2×#44739, #7668958, #7669114, … |
| maximal quotient of | #535821, #535896, #536387, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 37727);