Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| 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 | #2543 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
772 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)(3,7)(4,8)(5,6)(9,10,14,12)(11,15,13,16)(17,18,22,25)(19,20,23,28)(21,24,30,32)(26,27,29,31);(2,6)(3,8)(10,12)(11,13)(17,19,22,23)(18,20,25,28)(21,29,30,26)(24,31,32,27);(1,3)(2,4)(5,8)(6,7)(9,11,14,13)(10,15,12,16)(17,19)(18,20)(21,26)(22,23)(24,27)(25,28)(29,30)(31,32);(1,3)(2,4)(5,8)(6,7)(9,12,14,10)(11,15,13,16)(17,20)(18,23)(19,25)(21,27)(22,28)(24,29)(26,32)(30,31);(1,4)(2,3)(5,7)(6,8)(9,13,14,11)(10,15,12,16)(17,21)(18,24)(19,26)(20,27)(22,30)(23,29)(25,32)(28,31)Polycyclic presentation:
2^8|a^2=gh,b^2=fh,c^2=fg,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#56238, #8102711, #8103191, … |
| maximal normal subgroup of | C2×#56238, #8102711, #8103191, … |
| maximal quotient of | #585614, #586697, #587259, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49226);