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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1039 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(6,7)(9,10)(11,12)(13,14)(15,16);(1,2,5,3)(4,7,8,6)(10,14)(12,15)(17,18,20,22)(19,23,24,21);(9,11)(10,15)(12,14)(13,16)(17,18)(19,21)(20,22)(23,24);(1,3,5,2)(4,7,8,6)(9,11)(10,15)(12,14)(13,16);(1,4,5,8)(2,6,3,7)(9,12,13,15)(10,11,14,16)(17,19,20,24)(18,21,22,23)Polycyclic presentation:
2^8|b^2=fgh,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 | #6699901, #6699904, #6699964, … |
| maximal normal subgroup of | #6699901, #6699904, #6699964, … |
| maximal quotient of | #521640, #522275, #522948, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32234);