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 | |
762 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,6,8,4)(10,13)(14,16)(17,18,22,20)(19,25,26,23)(21,28,29,24)(27,30,32,31);(1,3,5,8)(2,4,7,6)(19,26)(21,29)(23,25)(24,28);(9,10)(11,14)(12,13)(15,16)(17,19,22,26)(18,23,20,25)(21,27,29,32)(24,30,28,31);(1,3)(2,6)(4,7)(5,8)(9,11,12,15)(10,14,13,16)(17,20,22,18)(19,23,26,25)(21,28,29,24)(27,30,32,31);(1,4,5,6)(2,3,7,8)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)Polycyclic presentation:
2^8|a^2=fh,b^2=f,c^2=h,d^2=g,e^2=f,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×#44481, #7668986, #7669099, … |
| maximal normal subgroup of | C2×#44481, #7668986, #7669099, … |
| maximal quotient of | #535653, #535712, #536281, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 37469);