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 | |
712 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(10,14)(15,16)(17,18,23,25)(19,26,20,21)(22,30,31,24)(27,29,28,32);(1,2)(3,5)(4,7)(6,8)(17,19,23,20)(18,21,25,26)(22,27,31,28)(24,29,30,32);(2,3)(6,7)(9,10)(11,15)(12,16)(13,14)(17,20,23,19)(18,21,25,26)(22,28,31,27)(24,29,30,32);(1,3,5,2)(4,7,8,6)(9,11,13,12)(10,15,14,16)(17,21)(18,19)(20,25)(22,29)(23,26)(24,27)(28,30)(31,32);(1,4)(2,6)(3,7)(5,8)(9,12,13,11)(10,16,14,15)(17,22)(18,24)(19,27)(20,28)(21,29)(23,31)(25,30)(26,32)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=f,d^2=gh,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×#50806, #7706424, #7706503, … |
| maximal normal subgroup of | C2×#50806, #7706424, #7706503, … |
| maximal quotient of | #560870, #561147, #561990, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 43794);