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 | C2×#127 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
928 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(6,7)(9,10)(11,13)(12,15)(14,16)(18,20)(22,23);(1,2,5,4)(3,7,8,6)(10,11)(14,15)(17,18,21,20)(19,22,24,23);(9,10,13,11)(12,14,16,15)(17,19,21,24)(18,22,20,23);(1,3,5,8)(2,6,4,7)(9,11,13,10)(12,15,16,14);(1,4,5,2)(3,7,8,6)(9,12,13,16)(10,14,11,15)(17,20,21,18)(19,23,24,22)Polycyclic presentation:
2^8|b^2=fg,c^2=fgh,d^2=h,e^2=f,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 | #7091168, #7091186, #7091231, … |
| maximal normal subgroup of | #7091168, #7091186, #7091231, … |
| maximal quotient of | #532114, #532121, #532245, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52950);