Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | Aut(G) order 65536 |
| 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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
677 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)(3,7)(4,6)(5,8)(9,10)(11,17)(12,20)(13,14)(15,18)(16,21)(19,22)(23,24)(25,26,30,28)(27,31,34,35)(29,36,37,32)(33,39,40,38);(1,2,4,6)(3,7,8,5)(10,14)(11,18)(16,20)(19,24)(25,27)(26,31)(28,35)(29,33)(30,34)(32,38)(36,39)(37,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,23)(21,24)(25,28,30,26)(27,31,34,35)(29,36,37,32)(33,38,40,39);(1,3)(2,5)(4,8)(6,7)(9,10)(11,17)(12,16)(13,14)(15,18)(19,23)(20,21)(22,24)(25,27)(26,31)(28,35)(29,33)(30,34)(32,38)(36,39)(37,40);(9,12,13,21)(10,16,14,20)(11,19,18,24)(15,22,17,23)(25,29,30,37)(26,32,28,36)(27,33,34,40)(31,38,35,39)Polycyclic presentation:
2^8|a^2=fh,b^2=fg,c^2=fh,e^2=fgh,b^a=bf,c^a=cg,d^a=dfg,e^a=efgh,c^b=ch,d^b=df,d^c=dfgh46 conjugacy classes — view the full character table.
| maximal subgroup of | #7203072, #7203076, #7203136, … |
| maximal normal subgroup of | #7203072, #7203076, #7203136, … |
| maximal quotient of | #582177, #582388, #582399, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46130);