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 | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
757 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)(10,13)(14,16)(17,18,23,25)(19,26,20,21)(22,30,31,24)(27,29,28,32);(1,3)(2,7)(4,6)(5,8)(17,19,23,20)(18,21,25,26)(22,27,31,28)(24,29,30,32);(2,8)(3,4)(9,10)(11,14)(12,13)(15,16)(17,20,23,19)(18,21,25,26)(22,28,31,27)(24,29,30,32);(1,4,6,3)(2,5,8,7)(9,11,12,15)(10,14,13,16)(17,21)(18,19)(20,25)(22,29)(23,26)(24,27)(28,30)(31,32);(1,5)(2,3)(4,8)(6,7)(17,22,23,31)(18,24,25,30)(19,27,20,28)(21,29,26,32)Polycyclic presentation:
2^8|a^2=fh,b^2=f,c^2=f,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×#48935, #7704745, #7704960, … |
| maximal normal subgroup of | C2×#48935, #7704745, #7704960, … |
| maximal quotient of | #561295, #561382, #562447, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 41923);