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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
762 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,3)(4,6,8,7)(9,10,12,14)(11,13,15,16)(17,18)(19,23)(20,24)(21,22);(1,2,5,3)(4,7,8,6)(18,22)(19,24);(2,3)(6,7)(9,10)(11,13)(12,14)(15,16)(17,19)(18,20)(21,24)(22,23);(1,3,5,2)(4,7,8,6)(9,11,12,15)(10,13,14,16)(17,20)(18,19)(21,23)(22,24);(1,4,5,8)(2,6,3,7)Polycyclic presentation:
2^8|a^2=fg,b^2=fh,d^2=fg,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 | #6699407, #6699410, #6699470, … |
| maximal normal subgroup of | #6699407, #6699410, #6699470, … |
| maximal quotient of | #544644, #545162, #545619, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35828);