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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
731 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)(3,7)(4,5)(6,8)(9,10,12,14)(11,13,15,16)(17,18,23,27)(19,28,21,24)(20,25,26,22)(29,30,31,32);(2,5)(6,7)(17,19,23,21)(18,24,27,28)(20,31,26,29)(22,32,25,30);(1,3)(2,6)(4,8)(5,7)(9,10)(11,13)(12,14)(15,16)(17,20,23,26)(18,25,27,22)(19,29,21,31)(24,32,28,30);(1,2)(3,6)(4,5)(7,8)(17,21,23,19)(18,24,27,28)(20,29,26,31)(22,32,25,30);(1,3)(2,6)(4,8)(5,7)(9,11,12,15)(10,13,14,16)(17,22,23,25)(18,26,27,20)(19,30,21,32)(24,31,28,29)Polycyclic presentation:
2^8|a^2=fg,b^2=h,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×#51493, #7707320, #7707344, … |
| maximal normal subgroup of | C2×#51493, #7707320, #7707344, … |
| maximal quotient of | #560776, #561146, #561895, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 44481);