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, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
749 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,6)(7,8)(17,18,21,22)(19,24,23,20);(9,10,13,16)(11,15,12,14)(18,22)(19,23);(1,2)(3,7)(4,8)(5,6)(9,11,13,12)(10,14,16,15)(17,19,21,23)(18,20,22,24);(1,3,5,4)(2,7,6,8)(17,20)(18,23)(19,22)(21,24);(1,4,5,3)(2,8,6,7)(9,12,13,11)(10,15,16,14)Polycyclic presentation:
2^8|a^2=f,b^2=fh,c^2=h,d^2=fg,e^2=gh,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 | #98051, #166939, #166940, … |
| maximal normal subgroup of | #98051, #166939, #166940, … |
| maximal quotient of | #521519, #522218, #522764, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32691);