Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60367 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | C2×#537 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1085 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)(3,6)(4,5)(7,8)(9,10)(11,19)(12,16)(13,17)(14,18)(15,20)(21,23)(22,24);(2,5)(6,8)(9,10)(11,15)(12,17)(13,16)(14,18)(19,20)(21,24)(22,23);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,19)(18,24);(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,24,19);(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,19,24,23)Polycyclic presentation:
2^8|c^2=f,d^2=h,e^2=h,g^2=h,b^a=bf,c^a=cg,e^a=eh,g^a=gh,d^b=dh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #246328, #246330, #246331, … |
| maximal normal subgroup of | #246328, #246330, #246331, … |
| maximal quotient of | #6370887, #6371275, #6371388, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53508);