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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
739 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(9,10,13,11)(12,15,16,14)(17,18)(19,25)(20,28)(21,22)(23,26)(24,29)(27,30)(31,32);(1,2)(3,4)(5,6)(7,8)(9,11,13,10)(12,15,16,14)(18,22)(19,26)(20,29)(30,31);(2,6)(3,7)(9,12,13,16)(10,14,11,15)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,31)(29,32);(1,3,5,7)(2,4,6,8)(17,18)(19,25)(20,24)(21,22)(23,26)(27,31)(28,29)(30,32);(1,4)(2,7)(3,6)(5,8)(17,20,21,29)(18,24,22,28)(19,27,26,32)(23,30,25,31)Polycyclic presentation:
2^8|a^2=fg,b^2=fg,c^2=fg,d^2=fh,e^2=f,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×#47759, #7701706, #7701714, … |
| maximal normal subgroup of | C2×#47759, #7701706, #7701714, … |
| maximal quotient of | #576697, #576729, #576755, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 40747);