Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| 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 | |
905 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(3,7)(6,8)(10,13)(11,15)(17,18)(19,25)(20,24)(21,22)(23,26)(27,31)(28,29)(30,32);(1,2)(3,6)(4,5)(7,8)(18,22)(19,26)(24,29)(27,32);(2,5)(6,8)(9,10,12,13)(11,14,15,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,26)(18,23,22,25)(20,27,28,32)(24,30,29,31);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,20,21,28)(18,24,22,29)(19,27,26,32)(23,30,25,31)Polycyclic presentation:
2^8|c^2=g,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=dfg,e^a=efgh,c^b=ch,d^b=df,d^c=dfgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#52404, #8092391, #8092416, … |
| maximal normal subgroup of | C2×#52404, #8092391, #8092416, … |
| maximal quotient of | #582253, #582487, #582488, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45392);