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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
796 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10)(11,15)(12,13)(14,16)(17,18,20,22)(19,23,24,21);(1,2)(3,7)(4,5)(6,8)(10,13)(14,15);(2,5)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,20,24)(18,21,22,23);(1,2)(3,6)(4,5)(7,8)(9,11,12,16)(10,14,13,15);(1,3,4,8)(2,6,5,7)Polycyclic presentation:
2^8|a^2=g,c^2=g,d^2=f,e^2=fh,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 | #6688549, #6688550, #6688565, … |
| maximal normal subgroup of | #6688549, #6688550, #6688565, … |
| maximal quotient of | #544569, #545111, #545571, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35481);