Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
846 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)(3,4)(5,7)(6,8)(9,10,14,12)(11,13,16,15)(17,18,21,22)(19,24,20,23);(1,3,5,8)(2,4,7,6)(9,11,14,16)(10,15,12,13)(18,22)(23,24);(1,3)(2,6)(4,7)(5,8)(9,10,14,12)(11,15,16,13)(17,18)(19,23)(20,24)(21,22);(1,3)(2,6)(4,7)(5,8)(9,12,14,10)(11,13,16,15)(17,19,21,20)(18,23,22,24);(1,4,5,6)(2,3,7,8)(9,13)(10,11)(12,16)(14,15)(17,20,21,19)(18,24,22,23)Polycyclic presentation:
2^8|a^2=f,b^2=fg,c^2=fgh,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #7091171, #7091189, #7091236, … |
| maximal normal subgroup of | #7091171, #7091189, #7091236, … |
| maximal quotient of | #531997, #532119, #532208, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52553);