Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60771 |
| 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 | |
1013 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)(9,10,14,19)(11,15,20,28)(12,17,23,31)(13,16,25,18)(21,27,33,38)(22,29,34,26)(24,30,36,32)(35,39,40,37);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,21)(13,22)(14,20)(16,29)(17,27)(18,26)(19,28)(23,33)(24,35)(25,34)(30,39)(31,38)(32,37)(36,40);(3,8)(5,7)(10,16)(11,20)(13,25)(15,26)(17,30)(18,19)(21,33)(24,36)(27,37)(28,29)(31,32)(38,39);(2,6)(5,7)(9,12)(10,17)(11,21)(13,24)(14,23)(15,27)(16,30)(18,32)(19,31)(20,33)(22,35)(25,36)(26,37)(28,38)(29,39)(34,40);(12,23)(17,31)(21,33)(24,36)(27,38)(30,32)(35,40)(37,39)Polycyclic presentation:
2^8|a^2=h,b^2=f,g^2=h,b^a=bf,c^a=cg,d^a=df,g^a=gh,c^b=cfh,g^c=gh,e^d=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | #152096, #249861, #249863, … |
| maximal normal subgroup of | #152096, #249861, #249863, … |
| maximal quotient of | #6374314, #6374469, #6374977, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53814);