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 | #2699 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
823 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,8)(3,7,4,5)(17,18)(19,22)(20,28)(21,24)(23,29)(25,27)(26,31)(30,32);(1,3)(2,7)(4,6)(5,8)(10,13)(11,15)(17,18,21,24)(19,22,27,25)(20,28,29,23)(26,31,32,30);(2,8)(3,4)(9,10,12,13)(11,14,15,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,29)(30,32);(1,4,6,3)(2,5,8,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32);(1,5)(2,3)(4,8)(6,7)(9,11)(10,14)(12,15)(13,16)(17,20,21,29)(18,23,24,28)(19,26,27,32)(22,30,25,31)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=fgh,d^2=g,e^2=f,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×#49760, #7705463, #7705669, … |
| maximal normal subgroup of | C2×#49760, #7705463, #7705669, … |
| maximal quotient of | #561174, #561231, #562258, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 42748);