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 | #2661 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
767 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,4)(6,7)(9,10)(11,15)(12,14)(13,16)(17,18,22,20)(19,23,25,27)(21,28,29,24)(26,32,31,30);(1,2,5,4)(3,7,8,6)(9,11,14,13)(10,15,12,16)(18,20)(19,25)(21,29)(30,32);(9,12,14,10)(11,15,13,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31);(1,3,5,8)(2,6,4,7)(9,13,14,11)(10,15,12,16)(17,20,22,18)(19,23,25,27)(21,28,29,24)(26,30,31,32);(1,4,5,2)(3,7,8,6)(17,21,22,29)(18,24,20,28)(19,26,25,31)(23,30,27,32)Polycyclic presentation:
2^8|a^2=gh,b^2=fh,c^2=g,d^2=fg,e^2=f,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×#55877, #8099886, #8100305, … |
| maximal normal subgroup of | C2×#55877, #8099886, #8100305, … |
| maximal quotient of | #585993, #586512, #586987, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 48865);