Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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×#392 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1171 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)(3,8)(4,7)(5,6)(17,18)(19,23)(20,22)(21,24);(2,6)(4,8)(9,10)(11,15)(12,13)(14,16)(17,18,20,22)(19,21,24,23);(2,6)(3,7)(10,13)(14,15)(19,24)(21,23);(1,3,5,7)(2,4,6,8)(9,10)(11,14)(12,13)(15,16)(17,19,20,24)(18,21,22,23);(1,4)(2,7)(3,6)(5,8)(9,11)(10,14)(12,16)(13,15)(17,18)(19,21)(20,22)(23,24)Polycyclic presentation:
2^8|b^2=fg,d^2=f,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #133202, #253303, #253305, … |
| maximal normal subgroup of | #133202, #253303, #253305, … |
| maximal quotient of | #544430, #545028, #545498, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 33530);