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 | #2673 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
842 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)(11,15)(12,13)(14,16)(17,18,22,25)(19,26,21,23)(20,28,30,24)(27,31,29,32);(2,5)(6,7)(10,13)(14,15)(17,19,22,21)(18,23,25,26)(20,29,30,27)(24,32,28,31);(9,11)(10,14)(12,16)(13,15)(17,20,22,30)(18,24,25,28)(19,27,21,29)(23,31,26,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,21,22,19)(18,23,25,26)(20,27,30,29)(24,32,28,31);(1,3,4,8)(2,6,5,7)(9,11)(10,14)(12,16)(13,15)(17,18)(19,26)(20,24)(21,23)(22,25)(27,32)(28,30)(29,31)Polycyclic presentation:
2^8|a^2=h,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×#51515, #7707385, #7707387, … |
| maximal normal subgroup of | C2×#51515, #7707385, #7707387, … |
| maximal quotient of | #560854, #561002, #561961, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 44503);