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×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
919 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)(4,8)(5,6)(9,10,12,14)(11,13,15,16)(17,18,21,22)(19,24,23,20);(2,6)(3,8)(9,10,12,14)(11,13,15,16)(18,22)(19,23);(1,3)(2,4)(5,8)(6,7)(9,10)(11,13)(12,14)(15,16)(17,19,21,23)(18,20,22,24);(1,3)(2,4)(5,8)(6,7)(9,10)(11,13)(12,14)(15,16)(17,20)(18,23)(19,22)(21,24);(1,4)(2,3)(5,7)(6,8)(9,11,12,15)(10,13,14,16)(17,20)(18,23)(19,22)(21,24)Polycyclic presentation:
2^8|a^2=fh,b^2=fg,c^2=gh,e^2=fg,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 | #7007778, #7007798, #7007831, … |
| maximal normal subgroup of | #7007778, #7007798, #7007831, … |
| maximal quotient of | #531886, #531941, #532140, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52759);