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 | C2×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
876 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,15)(14,16)(17,18,21,25)(19,24,27,22)(20,23,29,32)(26,31,28,30);(4,8)(6,7)(10,11)(12,16)(17,19,21,27)(18,22,25,24)(20,28,29,26)(23,31,32,30);(2,3)(6,7)(10,11)(14,15)(17,20,21,29)(18,23,25,32)(19,26,27,28)(22,30,24,31);(1,3,5,2)(4,7,8,6)(9,11,13,10)(12,15,16,14)(17,19)(18,24)(20,28)(21,27)(22,25)(23,30)(26,29)(31,32);(1,4)(2,6)(3,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,18)(19,24)(20,23)(21,25)(22,27)(26,31)(28,30)(29,32)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=h,d^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×#57603, #8101515, #8101947, … |
| maximal normal subgroup of | C2×#57603, #8101515, #8101947, … |
| maximal quotient of | #585229, #585880, #586861, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 50591);