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×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
912 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,5)(6,8)(10,14)(15,16)(17,18)(19,20)(21,23)(22,24);(2,5)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19,21,24)(18,20,23,22);(9,11,13,12)(10,15,14,16)(17,18,21,23)(19,22,24,20);(1,2)(3,6)(4,5)(7,8)(9,12,13,11)(10,16,14,15)(17,19)(18,22)(20,23)(21,24);(1,3)(2,6)(4,8)(5,7)(9,10)(11,15)(12,16)(13,14)(17,20,21,22)(18,19,23,24)Polycyclic presentation:
2^8|b^2=g,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 | #133646, #500480, #502771, … |
| maximal normal subgroup of | #133646, #500480, #502771, … |
| maximal quotient of | #532173, #532231, #532710, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53006);