Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| automorphism | Aut(G) order 688128 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 | |
1648 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,7)(3,4,8,6)(9,10,12,13)(11,15,16,14)(19,23)(22,24);(1,3,5,8)(2,6,7,4)(10,13)(14,15)(18,21)(22,24);(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,21)(23,24);(1,3)(2,4)(5,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,22)(20,23)(21,24);(1,4)(2,3)(5,6)(7,8)(9,11)(10,14)(12,16)(13,15)(17,18)(19,22)(20,21)(23,24)Polycyclic presentation:
2^8|a^2=f,b^2=fg,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.
| Fitting subgroup of | #53169⋊C3, #53169⋊C7 |
| maximal subgroup of | C2×#53169, #8105668, #8108057, … |
| maximal normal subgroup of | C2×#53169, #8105668, #8108057, … |
| maximal quotient of | #606711, #606885, #606898, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46157);