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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
771 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,3)(4,7,8,6)(9,10,14,15)(11,13,12,16);(2,3)(6,7)(10,15)(11,12)(17,18,21,24)(19,22,27,25)(20,28,29,23)(26,31,32,30);(9,11,14,12)(10,16,15,13)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,29)(30,32);(1,3,5,2)(4,7,8,6)(9,12,14,11)(10,13,15,16)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32);(1,4)(2,6)(3,7)(5,8)(9,13)(10,11)(12,15)(14,16)(17,20,21,29)(18,23,24,28)(19,26,27,32)(22,30,25,31)Polycyclic presentation:
2^8|a^2=f,b^2=gh,c^2=g,d^2=f,e^2=gh,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×#58951, #8105196, #8105208, … |
| maximal normal subgroup of | C2×#58951, #8105196, #8105208, … |
| maximal quotient of | #585582, #585887, #586293, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51939);