Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
1036 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,4)(3,7,8,6)(9,10,13,16)(11,12,15,14)(17,18,22,19)(20,23,21,24);(2,4)(6,7)(11,15)(12,14)(17,19,22,18)(20,23,21,24);(9,11,13,15)(10,14,16,12)(17,20,22,21)(18,23,19,24);(1,3,5,8)(2,6,4,7);(1,4,5,2)(3,7,8,6)(9,12,13,14)(10,15,16,11)(17,21,22,20)(18,24,19,23)Polycyclic presentation:
2^8|a^2=fh,b^2=gh,c^2=h,d^2=f,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 | #98303, #252901, #252904, … |
| maximal normal subgroup of | #98303, #252901, #252904, … |
| maximal quotient of | #521476, #522131, #522636, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32723);