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 | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
803 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10,13,14)(11,16,15,12)(19,23)(22,24);(1,2,4,6)(3,7,8,5)(10,14)(11,15)(17,18,21,20)(19,22,23,24);(1,2)(3,5)(4,6)(7,8)(9,11,13,15)(10,12,14,16)(17,19)(18,22)(20,24)(21,23);(9,12)(10,15)(11,14)(13,16)(17,20,21,18)(19,24,23,22);(1,3,4,8)(2,5,6,7)Polycyclic presentation:
2^8|a^2=f,b^2=gh,c^2=f,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 | #6700925, #6700928, #6700988, … |
| maximal normal subgroup of | #6700925, #6700928, #6700988, … |
| maximal quotient of | #521639, #522272, #522943, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32237);