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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
846 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)(4,5)(6,8)(17,18,22,21)(19,25,26,23)(20,28,29,24)(27,30,32,31);(2,5)(6,7)(9,10,13,11)(12,15,16,14)(19,26)(23,25)(27,32)(30,31);(10,11)(14,15)(17,19,22,26)(18,23,21,25)(20,27,29,32)(24,30,28,31);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,15,16,14)(17,20,22,29)(18,24,21,28)(19,27,26,32)(23,30,25,31);(1,3)(2,6)(4,8)(5,7)(9,12)(10,14)(11,15)(13,16)(17,21,22,18)(19,25,26,23)(20,28,29,24)(27,31,32,30)Polycyclic presentation:
2^8|a^2=g,b^2=gh,c^2=g,d^2=h,e^2=g,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×#58364, #8104095, #8104111, … |
| maximal normal subgroup of | C2×#58364, #8104095, #8104111, … |
| maximal quotient of | #585364, #586661, #587036, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51352);