Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| automorphism | Aut(G) order 65536 |
| 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 | |
1009 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,3)(4,7,8,6)(10,13)(11,15)(17,18)(19,20)(21,24)(22,23);(2,3)(6,7)(9,10,12,13)(11,14,15,16)(17,18,21,24)(19,22,23,20);(9,10)(11,14)(12,13)(15,16)(17,19,21,23)(18,22,24,20);(1,3,5,2)(4,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,23)(21,24);(1,4,5,8)(2,6,3,7)(9,11,12,15)(10,14,13,16)(17,20,21,22)(18,23,24,19)Polycyclic presentation:
2^8|a^2=fg,b^2=g,c^2=gh,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=df,e^b=egh46 conjugacy classes — view the full character table.
| maximal subgroup of | #98118, #179271, #179272, … |
| maximal normal subgroup of | #98118, #179271, #179272, … |
| maximal quotient of | #606803, #606817, #606819, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46363);