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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1033 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)(3,7)(4,5)(6,8)(9,10,12,13)(11,15,16,14)(18,22)(20,23);(2,5)(6,7)(10,13)(14,15)(17,18)(19,23)(20,24)(21,22);(9,10)(11,14)(12,13)(15,16)(17,19,21,24)(18,23,22,20);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16);(1,3,4,8)(2,6,5,7)(9,11,12,16)(10,14,13,15)(17,20,21,23)(18,24,22,19)Polycyclic presentation:
2^8|a^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=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #7011360, #7011380, #7011417, … |
| maximal normal subgroup of | #7011360, #7011380, #7011417, … |
| maximal quotient of | #531977, #532052, #532298, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52382);