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 | |
839 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)(3,7)(4,5)(6,8)(9,10)(11,13)(12,15)(14,16)(17,18,22,25)(19,20,23,28)(21,24,30,32)(26,27,29,31);(2,5)(6,7)(10,11)(14,15)(17,19,22,23)(18,20,25,28)(21,29,30,26)(24,31,32,27);(9,10,13,11)(12,14,16,15)(17,19)(18,20)(21,26)(22,23)(24,27)(25,28)(29,30)(31,32);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,15,16,14)(17,20)(18,23)(19,25)(21,27)(22,28)(24,29)(26,32)(30,31);(1,3,4,8)(2,6,5,7)(9,12,13,16)(10,14,11,15)(17,21)(18,24)(19,26)(20,27)(22,30)(23,29)(25,32)(28,31)Polycyclic presentation:
2^8|a^2=gh,b^2=gh,c^2=g,d^2=g,e^2=f,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×#55920, #8100675, #8101178, … |
| maximal normal subgroup of | C2×#55920, #8100675, #8101178, … |
| maximal quotient of | #585880, #585965, #586381, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 48908);