Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60457 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | #3500 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1019 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)(3,6)(4,5)(7,8)(9,10)(11,21)(12,16)(13,18)(14,29)(15,20)(17,24)(19,22)(23,35)(25,30)(26,31)(27,33)(28,39)(32,37)(34,36)(38,40);(2,5)(6,8)(9,11,22,26,15,24,14,12)(10,16,29,17,20,31,19,21)(13,23,36,38,27,37,28,25)(18,30,39,32,33,40,34,35);(1,3,4,7)(2,6,5,8)(9,12,14,24,15,26,22,11)(10,17,19,16,20,21,29,31)(13,25,28,37,27,38,36,23)(18,32,34,30,33,35,39,40);(9,13)(10,18)(11,23)(12,25)(14,28)(15,27)(16,30)(17,32)(19,34)(20,33)(21,35)(22,36)(24,37)(26,38)(29,39)(31,40);(13,27)(18,33)(23,37)(25,38)(28,36)(30,40)(32,35)(34,39)Polycyclic presentation:
2^8|b^2=gh,c^2=f,f^2=h,g^2=h,b^a=bf,c^a=cg,f^a=fh,g^a=gh,e^d=eh64 conjugacy classes — view the full character table.
| Fitting subgroup of | #60624⋊C3 |
| maximal subgroup of | #152075, #512640, #512641, … |
| maximal normal subgroup of | #152075, #512640, #512641, … |
| maximal quotient of | #6371056, #6371698, #6371699, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53612);