Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | Aut(G) order 98304 |
| 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 | |
805 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)(4,7,8,6)(17,18)(19,23)(20,21)(22,24);(2,3)(6,7)(9,10)(11,12)(13,14)(15,16)(18,21)(19,24);(10,14)(11,15)(17,18,20,21)(19,22,24,23);(1,3,5,2)(4,7,8,6)(9,11,13,15)(10,12,14,16)(17,18)(19,22)(20,21)(23,24);(1,4,5,8)(2,6,3,7)(9,12)(10,15)(11,14)(13,16)(17,19,20,24)(18,22,21,23)Polycyclic presentation:
2^8|a^2=fg,c^2=g,d^2=fh,e^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| Fitting subgroup of | #47733⋊C3 |
| maximal subgroup of | C2×#47733, #7701677, #7701710, … |
| maximal normal subgroup of | C2×#47733, #7701677, #7701710, … |
| maximal quotient of | #576756, #577226, #577770, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 40721);