Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60460 |
| 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 | #3512 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
921 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)(3,6)(4,5)(7,8)(9,10,14,18)(11,15,20,29)(12,22,24,39)(13,26,27,17)(16,31,33,25)(19,34,23,38)(21,37,36,30)(28,40,32,35);(2,5)(6,8)(9,11)(10,15)(12,23)(13,21)(14,20)(16,32)(17,30)(18,29)(19,24)(22,38)(25,40)(26,37)(27,36)(28,33)(31,35)(34,39);(9,12,13,25,14,24,27,31)(10,16,17,22,18,33,26,39)(11,19,21,35,20,23,36,40)(15,28,30,34,29,32,37,38);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,19)(13,21)(14,20)(16,28)(17,30)(18,29)(22,34)(23,24)(25,35)(26,37)(27,36)(31,40)(32,33)(38,39);(11,20)(12,24)(15,29)(16,33)(21,36)(22,39)(25,31)(30,37)Polycyclic presentation:
2^8|a^2=h,c^2=g,d^2=f,g^2=h,b^a=bf,c^a=cg,g^a=gh,c^b=ch,e^b=eh,e^c=eh,e^d=eh58 conjugacy classes — view the full character table.
| maximal subgroup of | #129982, #152092, #249831, … |
| maximal normal subgroup of | #129982, #152092, #249831, … |
| maximal quotient of | #6371193, #6372144, #6372462, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53714);