Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C25 |
| commutator | C22 | C24×C4 |
| prime radical | C24 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
987 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,10)(8,12)(9,11)(13,14)(15,19)(16,23)(17,18)(20,24)(21,27)(22,25)(26,28);(5,7,9,12)(6,10,11,8)(14,18)(19,25)(20,23)(26,27);(1,3,4,2)(5,7)(6,8)(9,12)(10,11)(13,15,17,22)(14,19,18,25)(16,21,24,28)(20,26,23,27);(5,8)(6,7)(9,10)(11,12)(13,14)(15,19)(16,20)(17,18)(21,26)(22,25)(23,24)(27,28);(13,16,17,24)(14,20,18,23)(15,21,22,28)(19,26,25,27)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df76 conjugacy classes — view the full character table.
| maximal subgroup of | #6388045, #6388047, #6388055, … |
| maximal normal subgroup of | #6388045, #6388047, #6388055, … |
| maximal quotient of | #513797, #513895, #513905, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27408);