Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62265 |
| 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 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
906 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)(11,18)(12,25)(13,21)(14,20)(15,19)(16,24)(17,28)(22,36)(23,31)(26,34)(27,33)(29,35)(30,40)(32,37)(38,39);(1,3)(2,5)(4,8)(6,7)(9,11,14,24)(10,16,20,18)(12,22,27,38)(13,29,15,23)(17,30,34,32)(19,35,21,31)(25,39,33,36)(26,40,28,37);(9,12,15,28,14,27,13,26)(10,17,21,25,20,34,19,33)(11,22,23,37,24,38,29,40)(16,30,31,39,18,32,35,36);(3,8)(5,7)(9,11)(10,18)(12,22)(13,29)(14,24)(15,23)(16,20)(17,32)(19,31)(21,35)(25,36)(26,40)(27,38)(28,37)(30,34)(33,39);(2,6)(5,7)(9,13,14,15)(10,19,20,21)(11,23,24,29)(12,26,27,28)(16,31,18,35)(17,33,34,25)(22,37,38,40)(30,39,32,36)Polycyclic presentation:
2^8|b^2=h,c^2=g,e^2=h,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=df,e^b=eh,e^d=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #130232, #130240, #130434, … |
| maximal normal subgroup of | #130232, #130240, #130434, … |
| maximal quotient of | #6386763, #6389167, #6389169, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55315);