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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
750 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,7)(3,8,5,4)(9,10,13,16)(11,14,12,15)(17,18,22,25)(19,21,24,30)(20,23,27,31)(26,29,28,32);(2,7)(3,5)(9,11,13,12)(10,14,16,15)(19,24)(20,27)(21,30)(23,31);(1,3,6,5)(2,8,7,4)(9,10)(11,14)(12,15)(13,16)(17,19,22,24)(18,21,25,30)(20,28,27,26)(23,32,31,29);(1,4,6,8)(2,3,7,5)(9,12,13,11)(10,15,16,14)(17,20)(18,23)(19,26)(21,29)(22,27)(24,28)(25,31)(30,32);(1,5,6,3)(2,8,7,4)(17,21)(18,24)(19,25)(20,29)(22,30)(23,28)(26,31)(27,32)Polycyclic presentation:
2^8|a^2=gh,b^2=g,c^2=h,d^2=fg,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×#50264, #7706086, #7706219, … |
| maximal normal subgroup of | C2×#50264, #7706086, #7706219, … |
| maximal quotient of | #561201, #561228, #562402, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 43252);