Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | #2543 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
890 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,7)(3,4,8,6)(9,10,14,12)(11,13,16,15)(17,18)(19,20)(21,23)(22,24);(1,3,5,8)(2,6,7,4)(9,11,14,16)(10,15,12,13)(17,19,21,24)(18,20,23,22);(9,10,14,12)(11,15,16,13)(17,18,21,23)(19,22,24,20);(1,3)(2,4)(5,8)(6,7)(9,12,14,10)(11,13,16,15)(17,19)(18,22)(20,23)(21,24);(1,4)(2,3)(5,6)(7,8)(9,13)(10,11)(12,16)(14,15)(17,20,21,22)(18,19,23,24)Polycyclic presentation:
2^8|a^2=f,b^2=fg,c^2=fh,d^2=fgh,e^2=g,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #130347, #486010, #486504, … |
| maximal normal subgroup of | #130347, #486010, #486504, … |
| maximal quotient of | #532173, #532232, #532712, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53007);