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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
758 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)(17,18)(19,21)(20,27)(22,24)(23,25)(26,29)(28,31)(30,32);(2,3)(6,7)(9,10)(11,12)(13,14)(15,16)(17,19,23,20)(18,21,25,27)(22,28,30,26)(24,31,32,29);(10,14)(11,15)(17,20,23,19)(18,21,25,27)(22,28,30,26)(24,29,32,31);(1,3,5,2)(4,7,8,6)(9,11,13,15)(10,12,14,16)(17,21)(18,20)(19,25)(22,29)(23,27)(24,28)(26,32)(30,31);(1,4)(2,6)(3,7)(5,8)(9,12)(10,15)(11,14)(13,16)(17,22,23,30)(18,24,25,32)(19,26,20,28)(21,29,27,31)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=g,d^2=fh,e^2=g,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×#58136, #8104654, #8104696, … |
| maximal normal subgroup of | C2×#58136, #8104654, #8104696, … |
| maximal quotient of | #585893, #586445, #586803, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51124);