Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | Aut(G) order 98304 |
| 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 | |
712 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(5,8)(9,10,14,12)(11,15,19,17)(13,20,21,16)(18,23,24,22)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,38)(37,40);(1,2,4,6)(3,5,7,8)(10,12)(15,17)(16,20)(22,23)(26,29)(27,33)(28,36)(39,40);(1,3)(2,5)(4,7)(6,8)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,27,30,33)(26,31,29,35)(28,37,36,34)(32,40,38,39);(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,23,24,22)(25,28,30,36)(26,32,29,38)(27,34,33,37)(31,39,35,40);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29,30,26)(27,35,33,31)(28,38,36,32)(34,40,37,39)Polycyclic presentation:
2^8|a^2=fh,b^2=g,c^2=f,d^2=f,e^2=h,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| Fitting subgroup of | #47824⋊C3 |
| maximal subgroup of | C2×#47824, #7701900, #7701905, … |
| maximal normal subgroup of | C2×#47824, #7701900, #7701905, … |
| maximal quotient of | #576642, #577232, #577997, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 40812);