Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61049 |
| 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 | #532 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
877 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,15)(12,17)(13,16)(14,18)(19,25)(20,27)(21,29)(22,26)(23,32)(24,30)(28,35)(31,36)(33,37)(34,38)(39,40);(1,3)(2,6)(4,7)(5,8)(9,11,14,21)(10,15,18,29)(12,20,23,34)(13,22,25,35)(16,26,19,28)(17,27,32,38)(24,33,36,40)(30,37,31,39);(1,4,5,2)(3,7,8,6)(10,16)(13,25)(15,26)(17,30)(18,19)(22,35)(24,36)(27,37)(28,29)(31,32)(33,40)(38,39);(3,8)(6,7)(9,12)(10,17)(11,20)(13,24)(14,23)(15,27)(16,30)(18,32)(19,31)(21,34)(22,33)(25,36)(26,37)(28,39)(29,38)(35,40);(10,18)(11,21)(12,23)(16,19)(22,35)(24,36)(27,38)(37,39)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=f,g^2=h,b^a=bf,c^a=cg,e^a=eh,g^a=gh,d^b=df,e^b=eh,g^c=gh,e^d=eh58 conjugacy classes — view the full character table.
| maximal subgroup of | #131097, #131098, #131916, … |
| maximal normal subgroup of | #131097, #131098, #131916, … |
| maximal quotient of | #6379325, #6379858, #6380088, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54274);