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×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
789 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,4)(6,7)(9,10,13,15)(11,12,16,14)(17,18,22,26)(19,23,27,30)(20,24,31,28)(21,25,32,29);(1,2,5,4)(3,7,8,6)(9,11)(10,14)(12,15)(13,16)(19,27)(21,32)(23,30)(25,29);(9,12,13,14)(10,11,15,16)(17,19,22,27)(18,23,26,30)(20,29,31,25)(21,28,32,24);(1,3,5,8)(2,6,4,7)(9,12)(10,11)(13,14)(15,16)(17,20)(18,24)(19,25)(21,30)(22,31)(23,32)(26,28)(27,29);(1,4,5,2)(3,7,8,6)(9,10)(11,12)(13,15)(14,16)(17,21,22,32)(18,25,26,29)(19,28,27,24)(20,30,31,23)Polycyclic presentation:
2^8|a^2=gh,b^2=fg,c^2=gh,d^2=fg,e^2=fgh,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×#58979, #8105184, #8105210, … |
| maximal normal subgroup of | C2×#58979, #8105184, #8105210, … |
| maximal quotient of | #585580, #585695, #586303, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51967);