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 | #2671 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
837 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)(4,6,8,7)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,30)(31,32);(1,2,5,3)(4,7,8,6)(9,10,13,14)(11,12,15,16)(18,22)(23,25)(24,28)(30,31);(2,3)(6,7)(10,14)(11,15)(17,19,21,27)(18,23,22,25)(20,26,29,32)(24,30,28,31);(1,3,5,2)(4,7,8,6)(9,11)(10,12)(13,15)(14,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32);(1,4)(2,6)(3,7)(5,8)(9,12)(10,11)(13,16)(14,15)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32)Polycyclic presentation:
2^8|a^2=fg,b^2=h,c^2=f,d^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×#47070, #7701035, #7701111, … |
| maximal normal subgroup of | C2×#47070, #7701035, #7701111, … |
| maximal quotient of | #576920, #576938, #577061, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 40058);