Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60457 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | C2×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
881 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10,13,15)(11,19,21,32)(12,17,23,28)(14,26,25,18)(16,29,27,22)(20,33,34,40)(24,37,36,31)(30,39,38,35);(1,3,4,8)(2,5,6,7)(10,15)(16,27)(17,28)(18,26)(19,32)(30,38)(31,37)(33,40);(9,11,14,22,13,21,25,29)(10,16,18,19,15,27,26,32)(12,20,24,35,23,34,36,39)(17,30,31,33,28,38,37,40);(9,12)(10,17)(11,20)(13,23)(14,24)(15,28)(16,30)(18,31)(19,33)(21,34)(22,35)(25,36)(26,37)(27,38)(29,39)(32,40);(12,23)(17,28)(20,34)(24,36)(30,38)(31,37)(33,40)(35,39)Polycyclic presentation:
2^8|a^2=f,b^2=fh,c^2=g,g^2=h,b^a=bf,c^a=cg,g^a=gh,e^d=eh64 conjugacy classes — view the full character table.
| Fitting subgroup of | #60666⋊C3 |
| maximal subgroup of | #130017, #130040, #483057, … |
| maximal normal subgroup of | #130017, #130040, #483057, … |
| maximal quotient of | #6371141, #6371875, #6371876, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53654);