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 | #2433 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(9,10,13,16)(11,12,15,14);(2,6)(4,7)(11,15)(12,14)(17,18,20,22)(19,23,24,21);(9,11,13,15)(10,14,16,12)(17,18)(19,21)(20,22)(23,24);(1,3)(2,7)(4,6)(5,8);(1,4,5,7)(2,3,6,8)(9,12,13,14)(10,15,16,11)(17,19,20,24)(18,21,22,23)Polycyclic presentation:
2^8|a^2=g,b^2=gh,c^2=g,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 | #6692745, #6692751, #6692789, … |
| maximal normal subgroup of | #6692745, #6692751, #6692789, … |
| maximal quotient of | #521626, #522252, #522898, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32186);