Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| automorphism | Aut(G) order 786432 |
| 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 | C26 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1635 conjugacy classes of subgroups; only the 455 normal subgroups are shown.
This group has 1635 conjugacy classes of subgroups — too many to draw. Only its 455 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,23)(22,24);(1,2)(3,7)(4,5)(6,8)(9,11,13,15)(10,12,14,16)(17,19)(18,22)(20,23)(21,24);(2,5)(6,7)(10,14)(11,15)(17,19,21,24)(18,22,23,20);(1,2)(3,6)(4,5)(7,8)(9,10)(11,12)(13,14)(15,16)(17,19)(18,20)(21,24)(22,23);(1,3)(2,6)(4,8)(5,7)(9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,22)(23,24)Polycyclic presentation:
2^8|b^2=fg,c^2=f,b^a=bf,c^a=cg,c^b=ch,e^b=eh,d^c=dh64 conjugacy classes — view the full character table.
| Fitting subgroup of | #35369⋊C3 |
| maximal subgroup of | #97554, #129482, #134213, … |
| maximal normal subgroup of | #97554, #129482, #134213, … |
| maximal quotient of | #515290, #515583, #515992, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28357);