Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| 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 | |
756 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)(3,7)(4,8)(5,6)(17,18,22,20)(19,23,27,25)(21,28,29,24)(26,31,32,30);(1,3,5,8)(2,4,6,7)(9,10,13,15)(11,12,16,14)(18,20)(23,25)(24,28)(30,31);(2,6)(4,7)(9,11,13,16)(10,14,15,12)(17,19,22,27)(18,23,20,25)(21,26,29,32)(24,30,28,31);(1,4)(2,3)(5,7)(6,8)(9,11)(10,12)(13,16)(14,15)(17,20,22,18)(19,25,27,23)(21,28,29,24)(26,31,32,30);(1,3)(2,4)(5,8)(6,7)(9,12)(10,11)(13,14)(15,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)Polycyclic presentation:
2^8|a^2=fg,b^2=gh,c^2=fgh,d^2=fg,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#57317, #8100497, #8101182, … |
| maximal normal subgroup of | C2×#57317, #8100497, #8101182, … |
| maximal quotient of | #586104, #586150, #586406, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 50305);