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 | #2678 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
855 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)(4,6)(5,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,25)(19,23,26,20)(21,24,29,31)(27,30,32,28);(1,2,4,6)(3,5,8,7)(9,10,12,14)(11,13,16,15)(19,26)(20,23)(21,29)(24,31);(9,11)(10,13)(12,16)(14,15)(17,19,22,26)(18,23,25,20)(21,27,29,32)(24,30,31,28);(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,20,22,23)(18,19,25,26)(21,28,29,30)(24,27,31,32);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,16)(14,15)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=h,d^2=h,e^2=fg,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×#51436, #7706953, #7707040, … |
| maximal normal subgroup of | C2×#51436, #7706953, #7707040, … |
| maximal quotient of | #560838, #561003, #561961, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 44424);