Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| 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×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
870 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)(9,10)(11,12)(13,14)(15,16)(17,18)(19,24)(20,22)(21,23);(1,2)(3,5)(4,7)(6,8)(10,14)(12,15)(17,19,21,20)(18,22,23,24);(2,3)(6,7)(9,11)(10,15)(12,14)(13,16)(17,20,21,19)(18,22,23,24);(1,3,5,2)(4,7,8,6)(9,12,13,15)(10,11,14,16)(17,18)(19,24)(20,22)(21,23);(1,4,5,8)(2,6,3,7)Polycyclic presentation:
2^8|a^2=h,b^2=fg,c^2=fg,d^2=g,e^2=h,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | #7085062, #7085076, #7085125, … |
| maximal normal subgroup of | #7085062, #7085076, #7085125, … |
| maximal quotient of | #535544, #535605, #536180, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38208);