Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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 | |
763 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,4,7,8)(11,15)(14,16)(17,18,22,25)(19,23,26,20)(21,24,29,31)(27,30,32,28);(2,6)(3,7)(9,10,13,12)(11,14,15,16)(19,26)(20,23)(21,29)(24,31);(9,11)(10,14)(12,16)(13,15)(17,19,22,26)(18,23,25,20)(21,27,29,32)(24,30,31,28);(1,3)(2,4)(5,7)(6,8)(17,20,22,23)(18,19,25,26)(21,28,29,30)(24,27,31,32);(1,4)(2,3)(5,8)(6,7)(9,12,13,10)(11,16,15,14)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)Polycyclic presentation:
2^8|a^2=fh,b^2=g,c^2=h,d^2=h,e^2=g,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#51445, #7706971, #7707039, … |
| maximal normal subgroup of | C2×#51445, #7706971, #7707039, … |
| maximal quotient of | #560840, #561145, #561959, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 44433);