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 | #2688 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
788 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(4,7)(9,10,13,12)(11,15,16,14)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,30)(31,32);(1,2,5,6)(3,4,8,7)(9,10,13,12)(11,14,16,15)(18,22)(23,25)(24,28)(30,31);(1,3,5,8)(2,7,6,4)(9,11,13,16)(10,14,12,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32);(1,3)(2,4)(5,8)(6,7)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32);(1,4)(2,3)(5,7)(6,8)(9,12,13,10)(11,15,16,14)(17,20,21,29)(18,24,22,28)(19,26,27,32)(23,30,25,31)Polycyclic presentation:
2^8|a^2=fgh,b^2=fg,c^2=fg,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×#49329, #7704307, #7704975, … |
| maximal normal subgroup of | C2×#49329, #7704307, #7704975, … |
| maximal quotient of | #560751, #560958, #561860, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 42317);