Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| 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 | #2670 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
799 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)(9,10,12,14)(11,15,16,13)(19,25)(23,28)(26,31)(30,32);(2,5)(6,7)(9,10,12,14)(11,13,16,15)(18,22)(20,27)(23,28)(26,31);(9,10)(11,13)(12,14)(15,16)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32);(1,2)(3,6)(4,5)(7,8)(9,11,12,16)(10,13,14,15)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32);(1,3,4,8)(2,6,5,7)(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=gh,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=df,e^b=egh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#53371, #8105852, #8105868, … |
| maximal normal subgroup of | C2×#53371, #8105852, #8105868, … |
| maximal quotient of | #606814, #606816, #606843, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46359);