Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| 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 | #2828 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
772 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)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,12,15)(18,19)(21,29)(23,25)(27,32);(2,4)(3,7)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30);(1,3,5,7)(2,6,4,8)(9,10)(11,14)(12,15)(13,16)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30);(1,3)(2,6)(4,8)(5,7)(9,11,13,12)(10,14,16,15)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31);(1,4,5,2)(3,8,7,6)(9,12,13,11)(10,15,16,14)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)Polycyclic presentation:
2^8|a^2=fgh,b^2=fh,c^2=f,d^2=g,e^2=fg,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#47417, #7701341, #7701440, … |
| maximal normal subgroup of | C2×#47417, #7701341, #7701440, … |
| maximal quotient of | #576963, #577010, #577079, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 40405);