Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4 | C25 |
| commutator, socle | C22 | C24×C4 |
| prime radical | #3455 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
893 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(3,7)(9,10,13,18,14,17,24,30)(11,15,21,26,20,27,33,37)(12,16,23,29,22,28,34,38)(19,25,32,36,31,35,39,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,19)(13,21)(14,20)(16,25)(17,27)(18,26)(22,31)(23,32)(24,33)(28,35)(29,36)(30,37)(34,39)(38,40);(2,5)(6,8)(9,12)(10,16)(11,19)(13,23)(14,22)(15,25)(17,28)(18,29)(20,31)(21,32)(24,34)(26,36)(27,35)(30,38)(33,39)(37,40);(1,3,4,7)(2,6,5,8)(12,22)(16,28)(19,31)(23,34)(25,35)(29,38)(32,39)(36,40);(10,17)(11,20)(16,28)(18,30)(19,31)(21,33)(29,38)(32,39)Polycyclic presentation:
2^8|a^2=g,d^2=f,g^2=h,b^a=bf,d^a=df,e^a=eh,c^b=cf,e^b=eh,d^c=dh76 conjugacy classes — view the full character table.
| maximal subgroup of | #129709, #129711, #209562, … |
| maximal normal subgroup of | #129709, #129711, #209562, … |
| maximal quotient of | #6370220, #6370227, #6370302, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53258);