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 | #2635 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
729 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10,14,13)(11,17,18,15)(12,20,21,16)(19,22,24,23)(26,27)(28,35)(29,37)(38,39);(4,8)(6,7)(11,18)(15,17)(19,24)(22,23)(25,26,30,27)(28,33,35,31)(29,34,37,32)(36,38,40,39);(2,3)(6,7)(9,11,14,18)(10,15,13,17)(12,19,21,24)(16,22,20,23)(25,27,30,26)(28,31,35,33)(29,34,37,32)(36,38,40,39);(1,3,5,2)(4,7,8,6)(9,12,14,21)(10,16,13,20)(11,19,18,24)(15,22,17,23)(25,28,30,35)(26,31,27,33)(29,36,37,40)(32,38,34,39);(1,4,5,8)(2,6,3,7)(9,13,14,10)(11,17,18,15)(12,20,21,16)(19,23,24,22)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=fh,d^2=g,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×#57016, #8099759, #8100306, … |
| maximal normal subgroup of | C2×#57016, #8099759, #8100306, … |
| maximal quotient of | #586411, #586615, #586831, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 50004);