Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| 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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
785 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,8)(3,4,7,6)(9,10)(11,15)(12,14)(13,16)(17,18,20,22)(19,23,24,21);(3,7)(4,6)(9,10,12,14)(11,13,16,15);(1,3,5,7)(2,6,8,4)(9,10)(11,13)(12,14)(15,16)(17,19,20,24)(18,21,22,23);(1,4,5,6)(2,7,8,3)(9,11)(10,13)(12,16)(14,15);(1,2)(3,6)(4,7)(5,8)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=g,d^2=fh,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #97258, #167929, #167930, … |
| maximal normal subgroup of | #97258, #167929, #167930, … |
| maximal quotient of | #553783, #554034, #554374, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30123);