Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| 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 | |
806 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,4)(3,7,8,6)(9,10,13,12)(11,15,16,14);(2,4)(6,7)(10,12)(14,15)(17,18,21,24)(19,23,20,22);(9,11)(10,14)(12,15)(13,16)(17,19,21,20)(18,22,24,23);(1,3,5,8)(2,6,4,7)(9,11)(10,14)(12,15)(13,16);(1,4,5,2)(3,7,8,6)(9,12,13,10)(11,15,16,14)(17,20,21,19)(18,23,24,22)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=h,d^2=fg,e^2=fh,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh58 conjugacy classes — view the full character table.
| maximal subgroup of | #6702444, #6702449, #6702509, … |
| maximal normal subgroup of | #6702444, #6702449, #6702509, … |
| maximal quotient of | #521512, #522213, #522773, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32649);