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 | |
892 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)(3,7)(4,6)(5,8)(9,10,13,18)(11,17,21,15)(12,23,25,39)(14,28,27,19)(16,31,33,26)(20,35,24,38)(22,30,37,34)(29,36,32,40);(1,3)(2,5)(4,8)(6,7)(9,11,13,21)(10,15,18,17)(12,24,25,20)(14,22,27,37)(16,32,33,29)(19,30,28,34)(23,35,39,38)(26,40,31,36);(9,12,14,26,13,25,27,31)(10,16,19,23,18,33,28,39)(11,20,22,36,21,24,37,40)(15,29,30,38,17,32,34,35);(3,8)(5,7)(9,11)(10,17)(12,24)(13,21)(14,22)(15,18)(16,29)(19,34)(20,25)(23,38)(26,40)(27,37)(28,30)(31,36)(32,33)(35,39);(2,6)(5,7)(10,18)(11,21)(16,33)(19,28)(20,24)(22,37)(23,39)(36,40)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=g,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^d=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #130447, #130449, #131738, … |
| maximal normal subgroup of | #130447, #130449, #131738, … |
| maximal quotient of | #6386769, #6389183, #6389327, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55329);