Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| 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 | #2428 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
961 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)(3,5)(4,7)(6,8)(18,21)(22,24);(1,3,6,4)(2,7,8,5)(9,10)(11,12)(13,14)(15,16)(17,18)(19,22)(20,21)(23,24);(2,8)(3,4)(10,14)(11,15);(1,4,6,3)(2,7,8,5)(9,11,13,15)(10,12,14,16);(1,5,6,7)(2,3,8,4)(9,12)(10,15)(11,14)(13,16)(17,19,20,23)(18,22,21,24)Polycyclic presentation:
2^8|b^2=g,d^2=h,e^2=f,b^a=bf,c^a=cg,c^b=ch,e^b=eh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #6455163, #6455165, #6455177, … |
| maximal normal subgroup of | #6455163, #6455165, #6455177, … |
| maximal quotient of | #515314, #515354, #515430, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28534);