Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| 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 | C2×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
748 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,14,12)(11,15,17,19)(13,20,21,16)(18,24,23,22)(25,26,30,29)(27,33,34,31)(28,36,37,32)(35,38,40,39);(1,2,4,6)(3,5,8,7)(10,12)(11,17)(13,21)(22,24)(27,34)(31,33)(35,40)(38,39);(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,27)(26,31)(28,35)(29,33)(30,34)(32,38)(36,39)(37,40);(1,2)(3,5)(4,6)(7,8)(9,12,14,10)(11,15,17,19)(13,20,21,16)(18,22,23,24)(25,28,30,37)(26,32,29,36)(27,35,34,40)(31,38,33,39);(1,3,4,8)(2,5,6,7)(9,13,14,21)(10,16,12,20)(11,18,17,23)(15,22,19,24)(25,29,30,26)(27,33,34,31)(28,36,37,32)(35,39,40,38)Polycyclic presentation:
2^8|a^2=h,b^2=fgh,d^2=h,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.
| Fitting subgroup of | #58460⋊C3 |
| maximal subgroup of | C2×#58460, #8104286, #8104459, … |
| maximal normal subgroup of | C2×#58460, #8104286, #8104459, … |
| maximal quotient of | #586492, #588548, #591854, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51448);